Package ‘tolerance’
February 17, 2016
Type Package
Title Functions for Calculating Tolerance Intervals
Version 1.2.0
Date 2016-02-17
Depends R (>= 3.2.2)
Imports rgl, stats4
Description Statistical tolerance limits provide the limits between which we can expect to find a specified proportion of a sampled population with a given level of confidence. This package provides functions for estimating tolerance limits for various distributions. Plotting is also available for tolerance limits of continuous random variables.
License GPL (>= 2)
NeedsCompilation no
Author Derek S. Young [aut, cre]
Maintainer Derek S. Young <derek.young@uky.edu>
Repository CRAN
Date/Publication 2016-02-17 17:42:19
R topics documented:
tolerance-package
acc.samp . . . . .
anovatol.int . . .
bayesnormtol.int
bintol.int . . . . .
bonftol.int . . . .
cautol.int . . . .
diffnormtol.int . .
DiffProp . . . . .
DiscretePareto . .
distfree.est . . . .
dpareto.ll . . . .
dparetotol.int . .
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R topics documented:
2
exp2tol.int . . . . .
exptol.int . . . . .
exttol.int . . . . . .
F1 . . . . . . . . .
fidbintol.int . . . .
fidnegbintol.int . .
fidpoistol.int . . . .
gamtol.int . . . . .
hypertol.int . . . .
K.factor . . . . . .
K.table . . . . . . .
laptol.int . . . . . .
logistol.int . . . . .
mvregtol.region . .
mvtol.region . . . .
negbintol.int . . . .
NegHypergeometric
neghypertol.int . .
nlregtol.int . . . . .
norm.OC . . . . .
norm.ss . . . . . .
normtol.int . . . .
np.order . . . . . .
npregtol.int . . . .
nptol.int . . . . . .
paretotol.int . . . .
plottol . . . . . . .
poislind.ll . . . . .
poislindtol.int . . .
PoissonLindley . .
poistol.int . . . . .
regtol.int . . . . . .
TwoParExponential
umatol.int . . . . .
uniftol.int . . . . .
ZipfMandelbrot . .
zipftol.int . . . . .
zm.ll . . . . . . . .
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tolerance-package
tolerance-package
3
Functions for Calculating Tolerance Intervals
Description
A collection of functions for calculating [100(1-alpha)%, 100(P)%] tolerance intervals, which
are intervals with 100(1-alpha)% confidence of covering 100(P)% of the sampled population of
interest.
Details
Package:
Type:
Version:
Date:
Imports:
License:
tolerance
Package
1.2.0
2016-02-17
rgl, stats4
GPL (>= 2)
Author(s)
Derek S. Young, Ph.D.
Maintainer: Derek S. Young <derek.young@uky.edu>
References
Hahn, G. J. and Meeker, W. Q. (1991), Statistical Intervals: A Guide for Practitioners, WileyInterscience.
Krishnamoorthy, K. and Mathew, T. (2009), Statistical Tolerance Regions: Theory, Applications,
and Computation, Wiley.
Patel, J. K. (1986), Tolerance Intervals - A Review, Communications in Statistics - Theory and
Methodology, 15, 2719–2762.
Young, D. S. (2010), tolerance: An R Package for Estimating Tolerance Intervals, Journal of
Statistical Software, 36(5), 1–39.
Young, D. S. (2014), Computing Tolerance Intervals and Regions in R. In M. B. Rao and C. R.
Rao (eds.), Handbook of Statistics, Volume 32: Computational Statistics with R, 309–338. NorthHolland, Amsterdam.
See Also
confint
4
acc.samp
acc.samp
Acceptance Sampling
Description
Provides an upper bound on the number of acceptable rejects or nonconformities in a process. This
is similar to a 1-sided upper tolerance bound for a hypergeometric random variable.
Usage
acc.samp(n, N, alpha = 0.05, P = 0.99, AQL = 0.01, RQL = 0.02)
Arguments
n
The sample size to be drawn from the inventory.
N
The total inventory (or lot) size.
alpha
1-alpha is the confidence level for bounding the probability of accepting the
inventory.
P
The proportion of items in the inventory which are to be accountable.
AQL
The acceptable quality level, which is the largest proportion of defects in a process considered acceptable. Note that 0 < AQL < 1.
RQL
The rejectable quality level, which is the largest proportion of defects in an
independent lot that one is willing to tolerate. Note that AQL < RQL < 1.
Value
acc.samp returns a matrix with the following quantities:
acceptance.limit
The number of items in the sample which may be unaccountable, yet still be
able to attain the desired confidence level 1-alpha.
lot.size
The total inventory (or lot) size N.
confidence
The confidence level 1-alpha.
P
The proportion of accountable items specified by the user.
AQL
The acceptable quality level as specified by the user. If the sampling were to be
repeated numerous times as a process, then this quantity specifies the proportion
of missing items considered acceptable from the process as a whole. Conditioning on the calculated value for acceptance.limit, the AQL is used to estimate
the producer’s risk (see prod.risk below).
RQL
The rejectable quality level as specified by the user. This is the proportion of
individual items in a sample one is willing to tolerate missing. Conditioning
on the calculated value for acceptance.limit, the RQL is used to estimate the
consumer’s risk (see cons.risk below).
sample.size
The sample size drawn as specified by n.
anovatol.int
5
prod.risk
The producer’s risk at the specified AQL. This is the probability of rejecting an
audit of a good inventory (also called the Type I error). A good inventory can be
rejected if an unfortunate random sample is selected (e.g., most of the missing
items happened to be selected for the audit). 1-prod.risk gives the confidence
level of this sampling plan for the specified AQL and RQL. If it is lower than
the confidence level desired (e.g., because the AQL is too high), then a warning
message will be displayed.
cons.risk
The consumer’s risk at the specified RQL. This is the probability of accepting
an audit of a bad inventory (also called the Type II error). A bad inventory can
be accepted if a fortunate random sample is selected (e.g., most of the missing
items happened to not be selected for the audit).
References
Montgomery, D. C. (2005), Introduction to Statistical Quality Control, Fifth Edition, John Wiley &
Sons, Inc.
See Also
Hypergeometric
Examples
## A 90%/90% acceptance sampling plan for a sample of 450
## drawn from a lot size of 960.
acc.samp(n = 450, N = 960, alpha = 0.10, P = 0.90, AQL = 0.07,
RQL = 0.10)
anovatol.int
Tolerance Intervals for ANOVA
Description
Tolerance intervals for each factor level in a balanced (or nearly-balanced) ANOVA.
Usage
anovatol.int(lm.out, data, alpha = 0.05, P = 0.99, side = 1,
method = c("HE", "HE2", "WBE", "ELL", "KM",
"EXACT", "OCT"), m = 50)
6
anovatol.int
Arguments
lm.out
An object of class lm (i.e., the results from the linear model fitting routine such
that the anova function can act upon).
data
A data frame consisting of the data fitted in lm.out. Note that data must have
one column for each main effect (i.e., factor) that is analyzed in lm.out and that
these columns must be of class factor.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
method
The method for calculating the k-factors. The k-factor for the 1-sided tolerance intervals is performed exactly and thus is the same for the chosen method.
"HE" is the Howe method and is often viewed as being extremely accurate, even
for small sample sizes. "HE2" is a second method due to Howe, which performs similarly to the Weissberg-Beatty method, but is computationally simpler. "WBE" is the Weissberg-Beatty method (also called the Wald-Wolfowitz
method), which performs similarly to the first Howe method for larger sample
sizes. "ELL" is the Ellison correction to the Weissberg-Beatty method when f is
appreciably larger than n^2. A warning message is displayed if f is not larger
than n^2. "KM" is the Krishnamoorthy-Mathew approximation to the exact solution, which works well for larger sample sizes. "EXACT" computes the k-factor
exactly by finding the integral solution to the problem via the integrate function. Note the computation time of this method is largely determined by m.
"OCT" is the Owen approach to compute the k-factor when controlling the tails
so that there is not more than (1-P)/2 of the data in each tail of the distribution.
m
The maximum number of subintervals to be used in the integrate function.
This is necessary only for method = "EXACT" and method = "OCT". The larger
the number, the more accurate the solution. Too low of a value can result in an error. A large value can also cause the function to be slow for method = "EXACT".
Value
anovatol.int returns a list where each element is a data frame corresponding to each main effect
(i.e., factor) tested in the ANOVA and the rows of each data frame are the levels of that factor. The
columns of each data frame report the following:
mean
The mean for that factor level.
n
The effective sample size for that factor level.
k
The k-factor for constructing the respective factor level’s tolerance interval.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
bayesnormtol.int
7
References
Howe, W. G. (1969), Two-Sided Tolerance Limits for Normal Populations - Some Improvements,
Journal of the American Statistical Association, 64, 610–620.
Weissberg, A. and Beatty, G. (1969), Tables of Tolerance Limit Factors for Normal Distributions,
Technometrics, 2, 483–500.
See Also
K.factor, normtol.int, lm, anova
Examples
## 90%/95% 2-sided tolerance intervals for a 2-way ANOVA
## using the "warpbreaks" data.
attach(warpbreaks)
lm.out <- lm(breaks ~ wool + tension)
out <- anovatol.int(lm.out, data = warpbreaks, alpha = 0.10,
P = 0.95, side = 2, method = "HE")
out
plottol(out, x = warpbreaks)
bayesnormtol.int
Bayesian Normal Tolerance Intervals
Description
Provides 1-sided or 2-sided Bayesian tolerance intervals under the conjugate prior for data distributed according to a normal distribution.
Usage
bayesnormtol.int(x = NULL, norm.stats = list(x.bar = NA,
s = NA, n = NA), alpha = 0.05, P = 0.99,
side = 1, method = c("HE", "HE2", "WBE",
"ELL", "KM", "EXACT", "OCT"), m = 50,
hyper.par = list(mu.0 = NULL,
sig2.0 = NULL, m.0 = NULL, n.0 = NULL))
Arguments
x
A vector of data which is distributed according to a normal distribution.
norm.stats
An optional list of statistics that can be provided in-lieu of the full dataset. If
provided, the user must specify all three components: the sample mean (x.bar),
the sample standard deviation (s), and the sample size (n).
8
bayesnormtol.int
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
method
The method for calculating the k-factors. The k-factor for the 1-sided tolerance intervals is performed exactly and thus is the same for the chosen method.
"HE" is the Howe method and is often viewed as being extremely accurate, even
for small sample sizes. "HE2" is a second method due to Howe, which performs similarly to the Weissberg-Beatty method, but is computationally simpler. "WBE" is the Weissberg-Beatty method (also called the Wald-Wolfowitz
method), which performs similarly to the first Howe method for larger sample
sizes. "ELL" is the Ellison correction to the Weissberg-Beatty method when f is
appreciably larger than n^2. A warning message is displayed if f is not larger
than n^2. "KM" is the Krishnamoorthy-Mathew approximation to the exact solution, which works well for larger sample sizes. "EXACT" computes the k-factor
exactly by finding the integral solution to the problem via the integrate function. Note the computation time of this method is largely determined by m.
"OCT" is the Owen approach to compute the k-factor when controlling the tails
so that there is not more than (1-P)/2 of the data in each tail of the distribution.
m
The maximum number of subintervals to be used in the integrate function.
This is necessary only for method = "EXACT" and method = "OCT". The larger
the number, the more accurate the solution. Too low of a value can result in an error. A large value can also cause the function to be slow for method = "EXACT".
hyper.par
A list consisting of the hyperparameters for the conjugate prior: the hyperparameters for the mean (mu.0 and n.0) and the hyperparameters for the variance
(sig2.0 and m.0).
Details
Note that if one considers the non-informative prior distribution, then the Bayesian tolerance intervals are the same as the classical solution, which can be obtained by using normtol.int.
Value
bayesnormtol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
x.bar
The sample mean.
1-sided.lower
The 1-sided lower Bayesian tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper Bayesian tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower Bayesian tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper Bayesian tolerance bound. This is given only if side = 2.
bintol.int
9
References
Aitchison, J. (1964), Bayesian Tolerance Regions, Journal of the Royal Statistical Society, Series
B, 26, 161–175.
Guttman, I. (1970), Statistical Tolerance Regions: Classical and Bayesian, Charles Griffin and
Company.
Young, D. S., Gordon, C. M., Zhu, S., and Olin, B. D. (2016), Sample Size Determination Strategies
for Normal Tolerance Intervals Using Historical Data, Quality Engineering (to appear).
See Also
Normal, normtol.int, K.factor
Examples
## 95%/85% 1-sided Bayesian normal tolerance limits for
## a sample of size 100.
set.seed(100)
x <- rnorm(100)
out <- bayesnormtol.int(x = x, alpha = 0.05, P = 0.85,
side = 1, method = "EXACT",
hyper.par = list(mu.0 = 0, sig2.0 = 1,
n.0 = 10, m.0 = 10))
out
plottol(out, x, plot.type = "both", side = "upper",
x.lab = "Normal Data")
bintol.int
Binomial Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for binomial random variables. From a statistical
quality control perspective, these limits use the proportion of defective (or acceptable) items in a
sample to bound the number of defective (or acceptable) items in future productions of a specified
quantity.
Usage
bintol.int(x, n, m = NULL, alpha = 0.05, P = 0.99, side = 1,
method = c("LS", "WS", "AC", "JF", "CP", "AS",
"LO", "PR", "PO", "CL", "CC", "CWS"),
a1 = 0.5, a2 = 0.5)
10
bintol.int
Arguments
x
The number of defective (or acceptable) units in the sample. Can be a vector of
length n, in which case the sum of x is used.
n
The size of the random sample of units selected for inspection.
m
The quantity produced in future groups. If m = NULL, then the tolerance limits
will be constructed assuming n for this quantity.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the defective (or acceptable) units in future samples of size m
to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
method
The method for calculating the lower and upper confidence bounds, which are
used in the calculation of the tolerance bounds. The default method is "LS",
which is the large-sample method. "WS" is Wilson’s method, which is just the
score confidence interval. "AC" gives the Agresti-Coull method, which is also
appropriate when the sample size is large. "JF" is Jeffreys’ method, which is
a Bayesian approach to the estimation. "CP" is the Clopper-Pearson (exact)
method, which is based on beta percentiles and provides a more conservative
interval. "AS" is the arcsine method, which is appropriate when the sample
proportion is not too close to 0 or 1. "LO" is the logit method, which also is
appropriate when the sample proportion is not too close to 0 or 1, but yields a
more conservative interval. "PR" uses a probit transformation and is accurate
for large sample sizes. "PO" is based on a Poisson parameterization, but it tends
to be more erratic compared to the other methods. "CL" is the complementary
log transformation and also tends to perform well for large sample sizes. "CC"
gives a continuity-corrected version of the large-sample method. "CWS" gives
a continuity-corrected version of Wilson’s method. More information on these
methods can be found in the "References".
a1
This specifies the first shape hyperparameter when using Jeffreys’ method.
a2
This specifies the second shape hyperparameter when using Jeffreys’ method.
Value
bintol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of defective (or acceptable) units in future samples of size m.
p.hat
The proportion of defective (or acceptable) units in the sample, calculated by
x/n.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
bintol.int
11
References
Brown, L. D., Cai, T. T., and DasGupta, A. (2001), Interval Estimation for a Binomial Proportion,
Statistical Science, 16, 101–133.
Hahn, G. J. and Chandra, R. (1981), Tolerance Intervals for Poisson and Binomial Variables, Journal of Quality Technology, 13, 100–110.
Newcombe, R. G. (1998), Two-Sided Confidence Intervals for the Single Proportion: Comparison
of Seven Methods, Statistics in Medicine, 17, 857–872.
See Also
Binomial, umatol.int
Examples
##
##
##
##
85%/90% 2-sided binomial tolerance intervals for a future
lot of 2500 when a sample of 230 were drawn from a lot of
1000. All methods but Jeffreys' method are compared
below.
bintol.int(x = 230, n = 1000,
side = 2, method =
bintol.int(x = 230, n = 1000,
side = 2, method =
bintol.int(x = 230, n = 1000,
side = 2, method =
bintol.int(x = 230, n = 1000,
side = 2, method =
bintol.int(x = 230, n = 1000,
side = 2, method =
bintol.int(x = 230, n = 1000,
side = 2, method =
bintol.int(x = 230, n = 1000,
side = 2, method =
bintol.int(x = 230, n = 1000,
side = 2, method =
bintol.int(x = 230, n = 1000,
side = 2, method =
bintol.int(x = 230, n = 1000,
side = 2, method =
bintol.int(x = 230, n = 1000,
side = 2, method =
##
##
##
##
##
##
m = 2500,
"LS")
m = 2500,
"WS")
m = 2500,
"AC")
m = 2500,
"CP")
m = 2500,
"AS")
m = 2500,
"LO")
m = 2500,
"PR")
m = 2500,
"PO")
m = 2500,
"CL")
m = 2500,
"CC")
m = 2500,
"CWS")
alpha = 0.15, P = 0.90,
alpha = 0.15, P = 0.90,
alpha = 0.15, P = 0.90,
alpha = 0.15, P = 0.90,
alpha = 0.15, P = 0.90,
alpha = 0.15, P = 0.90,
alpha = 0.15, P = 0.90,
alpha = 0.15, P = 0.90,
alpha = 0.15, P = 0.90,
alpha = 0.15, P = 0.90,
alpha = 0.15, P = 0.90,
Using Jeffreys' method to construct the 85%/90% 1-sided
binomial tolerance limits. The first calculation assumes
a prior on the proportion of defects which places greater
density on values near 0. The second calculation assumes
a prior on the proportion of defects which places greater
density on values near 1.
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
12
bonftol.int
side = 1, method = "JF", a1 = 2, a2 = 10)
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
side = 1, method = "JF", a1 = 5, a2 = 1)
bonftol.int
Approximate 2-Sided Tolerance Intervals that Control the Tails Using
Bonferroni’s Inequality
Description
This function allows the user to control what proportion of the population is to be in the tails of the
given distribution for a 2-sided tolerance interval. The result is a conservative approximation based
on Bonferroni’s inequality.
Usage
bonftol.int(fn, P1 = 0.005, P2 = 0.005, alpha = 0.05, ...)
Arguments
fn
The function name for the 2-sided tolerance interval to be calculated.
P1
The proportion of the population not covered in the lower tail of the distribution.
P2
The proportion of the population not covered in the upper tail of the distribution.
alpha
The level chosen such that 1-alpha is the confidence level.
...
Additional arguments passed to fn, including the data. All arguments that would
be specified in fn must also be specified here.
Value
The results for the 2-sided tolerance interval procedure are reported. See the corresponding help file
for fn about specific output. Note that the (minimum) proportion of the population to be covered
by this interval is 1 - (P1 + P2).
Note
This function can be used with any 2-sided tolerance interval function, including the regression
tolerance interval functions.
References
Jensen, W. A. (2009), Approximations of Tolerance Intervals for Normally Distributed Data, Quality and Reliability Engineering International, 25, 571–580.
Patel, J. K. (1986), Tolerance Intervals - A Review, Communications in Statistics - Theory and
Methodology, 15, 2719–2762.
cautol.int
13
Examples
## 95%/97% tolerance interval for normally distributed
## data controlling 1% of the data is in the lower tail
## and 2% of the data in the upper tail.
set.seed(100)
x <- rnorm(100, 0, 0.2)
bonftol.int(normtol.int, x = x, P1 = 0.01, P2 = 0.02,
alpha = 0.05, method = "HE")
cautol.int
Cauchy Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for Cauchy distributed data.
Usage
cautol.int(x, alpha = 0.05, P = 0.99, side = 1)
Arguments
x
A vector of data which is Cauchy distributed.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
Value
cautol.int returns a data.frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Bain, L. J. (1978), Statistical Analysis of Reliability and Life-Testing Models, Marcel Dekker, Inc.
14
diffnormtol.int
See Also
Cauchy
Examples
## 95%/90% 2-sided Cauchy tolerance interval for a sample
## of size 1000.
set.seed(100)
x <- rcauchy(1000, 100000, 10)
out <- cautol.int(x = x, alpha = 0.05, P = 0.90, side = 2)
out
plottol(out, x, plot.type = "both", x.lab = "Cauchy Data")
diffnormtol.int
1-Sided Tolerance Limits for the Distribution of the Difference Between Two Independent Normal Random Variables
Description
Provides 1-sided tolerance limits for the difference between two independent normal random variables. If the ratio of the variances is known, then an exact calculation is performed. Otherwise,
approximation methods are implemented.
Usage
diffnormtol.int(x1, x2, var.ratio = NULL, alpha = 0.05,
P = 0.99, method = c("HALL", "GK", "RG"))
Arguments
x1
A vector of sample data which is distributed according to a normal distribution
(sample 1).
x2
Another vector of sample data which is distributed according to a normal distribution (sample 2). It can be of a different sample size than the sample specified
by x1.
var.ratio
A specified, known value of the variance ratio (i.e., the ratio of the variance
for population 1 to the variance of population 2). If NULL, then the variance
ratio is estimated according to one of the three methods specified in the method
argument.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by the tolerance limits.
diffnormtol.int
method
15
The method for estimating the variance ratio. This only needs to be specified
in the case when var.ratio is not NULL. "HALL" is Hall’s method, which takes
a bias-corrected version of the ratio between the sample variance for sample 1
to the sample variance for sample 2. "GK" is the Guo-Krishnamoorthy method,
which first calculates a bias-corrected version of the ratio between the sample
variance for sample 2 to the sample variance for sample 1. The resulting limit is
then compared to the limit from Hall’s method and the most conservative limit
is chosen. "RG" is the Reiser-Guttman method, which is a biased version of
the variance ratio that is calculated by taking the sample variance for sample
1 to the sample variance for sample 2. Typically, Hall’s method or the GuoKrishnamoorthy method are preferred to the Reiser-Guttman method.
Details
Satterthwaite’s approximation for the degrees of freedom is used when the variance ratio is unknown.
Value
diffnormtol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
diff.bar
The difference between the sample means.
1-sided.lower
The 1-sided lower tolerance bound.
1-sided.upper
The 1-sided upper tolerance bound.
Note
Unlike other tolerance interval functions, the output from diffnormtol.int cannot be passed to
plottol.
References
Guo, H. and Krishnamoorthy, K. (2004), New Approximate Inferential Methods for the Reliability
Parameter in a Stress-Strength Model: The Normal Case, Communications in Statistics - Theory
and Methods, 33, 1715–1731.
Hall, I. J. (1984), Approximate One-Sided Tolerance Limits for the Difference or Sum of Two
Independent Normal Variates, Journal of Quality Technology, 16, 15–19.
Krishnamoorthy, K. and Mathew, T. (2009), Statistical Tolerance Regions: Theory, Applications,
and Computation, Wiley.
Reiser, B. J. and Guttman, I. (1986), Statistical Inference for Pr(Y < X): The Normal Case, Technometrics, 28, 253–257.
See Also
Normal, K.factor, normtol.int
16
DiffProp
Examples
##
##
##
##
##
90%/99% tolerance limits for the difference between two
simulated normal data sets. This data is taken from
Krishnamoorthy and Mathew (2009). Note that there is a
calculational error in their example, which yields different
results with the output below.
x1 <- c(10.166, 5.889, 8.258, 7.303, 8.757)
x2 <- c(-0.204, 2.578, 1.182, 1.892, 0.786, -0.517, 1.156,
0.980, 0.323, 0.437, 0.397, 0.050, 0.812, 0.720)
diffnormtol.int(x1,
diffnormtol.int(x1,
diffnormtol.int(x1,
diffnormtol.int(x1,
DiffProp
x2,
x2,
x2,
x2,
alpha = 0.10, P = 0.99, method
alpha = 0.10, P = 0.99, method
alpha = 0.10, P = 0.99, method
var.ratio = 3.8, alpha = 0.10,
=
=
=
P
"HALL")
"GK")
"RG")
= 0.99)
Difference Between Two Proportions Distribution
Description
Density (mass), distribution function, quantile function, and random generation for the difference
between two proportions. This is determined by taking the difference between two independent
beta distributions.
Usage
ddiffprop(x, k1, k2, n1, n2,
log = FALSE, ...)
pdiffprop(q, k1, k2, n1, n2,
lower.tail = TRUE,
qdiffprop(p, k1, k2, n1, n2,
lower.tail = TRUE,
rdiffprop(n, k1, k2, n1, n2,
a1 = 0.5, a2 = 0.5,
a1 = 0.5, a2 =
log.p = FALSE,
a1 = 0.5, a2 =
log.p = FALSE,
a1 = 0.5, a2 =
0.5,
...)
0.5,
...)
0.5)
Arguments
x, q
Vector of quantiles.
p
Vector of probabilities.
n
The number of observations. If length>1, then the length is taken to be the
number required.
k1, k2
The number of successes drawn from groups 1 and 2, respectively.
n1, n2
The sample sizes for groups 1 and 2, respectively.
DiffProp
17
a1, a2
The shift parameters for the beta distributions. For the fiducial approach, we
know that the lower and upper limits are set at a1 = a2 = 0 and a1 = a2 = 1,
respectively, for the true p1 and p2. While computations can be performed on
real values outside the unit interval, a warning message will be returned if such
values are specified. For practical purposes, the default value of 0.5 should be
used for each parameter.
log, log.p
Logical vectors. If TRUE, then the probabilities are given as log(p).
lower.tail
Logical vector. If TRUE, then probabilities are P [X ≤ x], else P [X > x].
...
Additional arguments passed to the Appell F1 function.
Details
The difference between two proportions distribution has a fairly complicated functional form. Please
see the article by Chen and Luo (2011), who corrected a typo in the article by Nadarajah and Kotz
(2007), for the functional form of this distribution.
Value
ddiffprop gives the density (mass), pdiffprop gives the distribution function, qdiffprop gives
the quantile function, and rdiffprop generates random deviates.
References
Chen, Y. and Luo, S. (2011), A Few Remarks on ’Statistical Distribution of the Difference of Two
Proportions’, Statistics in Medicine, 30, 1913–1915.
Nadarajah, S. and Kotz, S. (2007), Statistical Distribution of the Difference of Two Proportions,
Statistics in Medicine, 26, 3518–3523.
See Also
runif and .Random.seed about random number generation.
Examples
## Randomly generated data from the difference between
## two proportions distribution.
set.seed(100)
x <- rdiffprop(n = 100, k1 = 2, k2 = 10, n1 = 17, n2 = 13)
hist(x, main = "Randomly Generated Data", prob = TRUE)
x.1 <- sort(x)
y <- ddiffprop(x = x.1, k1 = 2, k2 = 10, n1 = 17, n2 = 13)
lines(x.1, y, col = 2, lwd = 2)
plot(x.1, pdiffprop(q = x.1, k1 = 2, k2 = 10, n1 = 17,
n2 = 13), type = "l", xlab = "x",
ylab = "Cumulative Probabilities")
qdiffprop(p = 0.20, k1 = 2, k2 = 10, n1 = 17, n2 = 13,
18
DiscretePareto
lower.tail = FALSE)
qdiffprop(p = 0.80, k1 = 2, k2 = 10, n1 = 17, n2 = 13)
DiscretePareto
Discrete Pareto Distribution
Description
Density (mass), distribution function, quantile function, and random generation for the discrete
Pareto distribution.
Usage
ddpareto(x,
pdpareto(q,
qdpareto(p,
rdpareto(n,
theta, log = FALSE)
theta, lower.tail = TRUE, log.p = FALSE)
theta, lower.tail = TRUE, log.p = FALSE)
theta)
Arguments
x, q
Vector of quantiles.
p
Vector of probabilities.
n
The number of observations. If length>1, then the length is taken to be the
number required.
theta
The shape parameter, which must be greater than 0 and less than 1.
log, log.p
Logical vectors. If TRUE, then the probabilities are given as log(p).
lower.tail
Logical vector. If TRUE, then probabilities are P [X ≤ x], else P [X > x].
Details
The discrete Pareto distribution has mass
p(x) = θlog(1+x) − θlog(2+x) ,
where x = 0, 1, . . . and 0 < θ < 1 is the shape parameter.
Value
ddpareto gives the density (mass), pdpareto gives the distribution function, qdpareto gives the
quantile function, and rdpareto generates random deviates for the specified distribution.
References
Krishna, H. and Pundir, P. S. (2009), Discrete Burr and Discrete Pareto Distributions, Statistical
Methodology, 6, 177–188.
Young, D. S., Naghizadeh Qomi, M., and Kiapour, A. (2016), Approximate Discrete Pareto Tolerance Limits for Characterizing Extremes in Count Data, submitted.
distfree.est
19
See Also
runif and .Random.seed about random number generation.
Examples
## Randomly generated data from the discrete Pareto
## distribution.
set.seed(100)
x <- rdpareto(n = 150, theta = 0.2)
hist(x, main = "Randomly Generated Data", prob = TRUE)
x.1 <- sort(x)
y <- ddpareto(x = x.1, theta = 0.2)
lines(x.1, y, col = 2, lwd = 2)
plot(x.1, pdpareto(q = x.1, theta = 0.2), type = "l",
xlab = "x", ylab = "Cumulative Probabilities")
qdpareto(p = 0.80, theta = 0.2, lower.tail = FALSE)
qdpareto(p = 0.95, theta = 0.2)
distfree.est
Estimating Various Quantities for Distribution-Free Tolerance Intervals
Description
When providing two of the three quantities n, alpha, and P, this function solves for the third quantity
in the context of distribution-free tolerance intervals.
Usage
distfree.est(n = NULL, alpha = NULL, P = NULL, side = 1)
Arguments
n
The necessary sample size to cover a proportion P of the population with confidence 1-alpha. Can be a vector.
alpha
1 minus the confidence level attained when it is desired to cover a proportion P
of the population and a sample size n is provided. Can be a vector.
P
The proportion of the population to be covered with confidence 1-alpha when
a sample size n is provided. Can be a vector.
side
Whether a 1-sided or 2-sided tolerance interval is assumed (determined by side = 1
or side = 2, respectively).
20
dpareto.ll
Value
When providing two of the three quantities n, alpha, and P, distfree.est returns the third quantity. If more than one value of a certain quantity is specified, then a table will be returned.
References
Natrella, M. G. (1963), Experimental Statistics: National Bureau of Standards - Handbook No. 91,
United States Government Printing Office, Washington, D.C.
See Also
nptol.int
Examples
## Solving for 1 minus the confidence level.
distfree.est(n = 59, P = 0.95, side = 1)
## Solving for the sample size.
distfree.est(alpha = 0.05, P = 0.95, side = 1)
## Solving for the proportion of the population to cover.
distfree.est(n = 59, alpha = 0.05, side = 1)
## Solving for sample sizes for many tolerance specifications.
distfree.est(alpha = seq(0.01, 0.05, 0.01),
P = seq(0.80, 0.99, 0.01), side = 2)
dpareto.ll
Maximum Likelihood Estimation for the Discrete Pareto Distribution
Description
Performs maximum likelihood estimation for the parameter of the discrete Pareto distribution.
Usage
dpareto.ll(x, theta = NULL, ...)
dparetotol.int
21
Arguments
x
A vector of raw data which is distributed according to a Poisson-Lindley distribution.
theta
Optional starting value for the parameter. If NULL, then the method of moments
estimator is used.
...
Additional arguments passed to the mle function.
Details
The discrete Pareto distribution is a discretized of the continuous Type II Pareto distribution (also
called the Lomax distribution).
Value
See the help file for mle to see how the output is structured.
References
Krishna, H. and Pundir, P. S. (2009), Discrete Burr and Discrete Pareto Distributions, Statistical
Methodology, 6, 177–188.
Young, D. S., Naghizadeh Qomi, M., and Kiapour, A. (2016), Approximate Discrete Pareto Tolerance Limits for Characterizing Extremes in Count Data, submitted.
See Also
mle, DiscretePareto
Examples
## Maximum likelihood estimation for randomly generated data
## from the discrete Pareto distribution.
set.seed(100)
dp.data <- rdpareto(n = 500, theta = 0.2)
out.dp <- dpareto.ll(dp.data)
stats4::coef(out.dp)
stats4::vcov(out.dp)
dparetotol.int
Discrete Pareto Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for data distributed according to the discrete Pareto
distribution.
22
dparetotol.int
Usage
dparetotol.int(x, m = NULL, alpha = 0.05, P = 0.99, side = 1,
...)
Arguments
x
A vector of raw data which is distributed according to a discrete Pareto distribution.
m
The number of observations in a future sample for which the tolerance limits
will be calculated. By default, m = NULL and, thus, m will be set equal to the
original sample size.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
...
Additional arguments passed to the dpareto.ll function, which is used for
maximum likelihood estimation.
Details
The discrete Pareto is a discretized of the continuous Type II Pareto distribution (also called the
Lomax distribution). Discrete Pareto distributions are heavily right-skewed distributions and potentially good models for discrete lifetime data and extremes in count data. For most practical
applications, one will typically be interested in 1-sided upper bounds.
Value
dparetotol.int returns a data frame with the following items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
theta
MLE for the shape parameter theta.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Young, D. S., Naghizadeh Qomi, M., and Kiapour, A. (2016), Approximate Discrete Pareto Tolerance Limits for Characterizing Extremes in Count Data, submitted.
See Also
DiscretePareto, dpareto.ll
exp2tol.int
23
Examples
## 95%/95% 1-sided tolerance intervals for data assuming
## the discrete Pareto distribution.
set.seed(100)
x <- rdpareto(n = 500, theta = 0.5)
out <- dparetotol.int(x, alpha = 0.05, P = 0.95, side = 1)
out
exp2tol.int
2-Parameter Exponential Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for data distributed according to a 2-parameter exponential distribution. Data with Type II censoring is permitted.
Usage
exp2tol.int(x, alpha = 0.05, P = 0.99, side = 1,
method = c("GPU", "DUN", "KM"), type.2 = FALSE)
Arguments
x
A vector of data which is distributed according to the 2-parameter exponential
distribution.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
method
The method for how the upper tolerance bound is approximated. "GPU" is the
Guenther-Patil-Upppuluri method. "DUN" is the Dunsmore method, which has
been empirically shown to be an improvement for samples greater than or equal
to 8. "KM" is the Krishnamoorthy-Mathew method, which is typically more
liberal than the other methods. More information on these methods can be found
in the "References", which also highlight general sample size conditions as to
when these different methods should be used.
type.2
Select TRUE if Type II censoring is present (i.e., the data set is censored at the
maximum value present). The default is FALSE.
24
exp2tol.int
Value
exp2tol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Dunsmore, I. R. (1978), Some Approximations for Tolerance Factors for the Two Parameter Exponential Distribution, Technometrics, 20, 317–318.
Engelhardt, M. and Bain, L. J. (1978), Tolerance Limits and Confidence Limits on Reliability for
the Two-Parameter Exponential Distribution, Technometrics, 20, 37–39.
Guenther, W. C., Patil, S. A., and Uppuluri, V. R. R. (1976), One-Sided β-Content Tolerance Factors
for the Two Parameter Exponential Distribution, Technometrics, 18, 333–340.
Krishnamoorthy, K. and Mathew, T. (2009), Statistical Tolerance Regions: Theory, Applications,
and Computation, Wiley.
See Also
TwoParExponential
Examples
## 95%/90% 1-sided 2-parameter exponential tolerance intervals
## for a sample of size 50.
set.seed(100)
x <- r2exp(50, 6, shift = 55)
out <- exp2tol.int(x = x, alpha = 0.05, P = 0.90, side = 1,
method = "DUN", type.2 = FALSE)
out
plottol(out, x, plot.type = "both", side = "upper",
x.lab = "2-Parameter Exponential Data")
exptol.int
exptol.int
25
Exponential Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for data distributed according to an exponential distribution. Data with Type II censoring is permitted.
Usage
exptol.int(x, alpha = 0.05, P = 0.99, side = 1, type.2 = FALSE)
Arguments
x
A vector of data which is distributed according to an exponential distribution.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
type.2
Select TRUE if Type II censoring is present (i.e., the data set is censored at the
maximum value present). The default is FALSE.
Value
exptol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
lambda.hat
The mean of the data (i.e., 1/rate).
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Blischke, W. R. and Murthy, D. N. P. (2000), Reliability: Modeling, Prediction, and Optimization,
John Wiley & Sons, Inc.
See Also
Exponential
26
exttol.int
Examples
## 95%/99% 1-sided exponential tolerance intervals for a
## sample of size 50.
set.seed(100)
x <- rexp(100, 0.004)
out <- exptol.int(x = x, alpha = 0.05, P = 0.99, side = 1,
type.2 = FALSE)
out
plottol(out, x, plot.type = "both", side = "lower",
x.lab = "Exponential Data")
exttol.int
Weibull (or Extreme-Value) Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for data distributed according to either a Weibull
distribution or extreme-value (also called Gumbel) distributions.
Usage
exttol.int(x, alpha = 0.05, P = 0.99, side = 1,
dist = c("Weibull", "Gumbel"), ext = c("min", "max"),
NR.delta = 1e-8)
Arguments
x
A vector of data which is distributed according to either a Weibull distribution
or an extreme-value distribution.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
dist
Select either dist = "Weibull" or dist = "Gumbel" if the data is distributed
according to the Weibull or extreme-value distribution, respectively.
ext
If dist = "Gumbel", then select which extreme is to be modeled for the Gumbel
distribution. The Gumbel distribution for the minimum (i.e., ext = "min")
corresponds to a left-skewed distribution and the Gumbel distribution for the
maximum (i.e., ext = "max") corresponds to a right-skewed distribution
NR.delta
The stopping criterion used for the Newton-Raphson algorithm when finding the
maximum likelihood estimates of the Weibull or extreme-value distribution.
exttol.int
27
Details
Recall that the relationship between the Weibull distribution and the extreme-value distribution for
the minimum is that if the random variable X is distributed according to a Weibull distribution, then
the random variable Y = ln(X) is distributed according to an extreme-value distribution for the
minimum.
If dist = "Weibull", then the natural logarithm of the data are taken so that a Newton-Raphson
algorithm can be employed to find the MLEs of the extreme-value distribution for the minimum
and then the data and MLEs are transformed back appropriately. No transformation is performed
if dist = "Gumbel". The Newton-Raphson algorithm is initialized by the method of moments
estimators for the parameters.
Value
exttol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
shape.1
MLE for the shape parameter if dist = "Weibull" or for the location parameter
if dist = "Gumbel".
shape.2
MLE for the scale parameter if dist = "Weibull" or dist = "Gumbel".
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Bain, L. J. and Engelhardt, M. (1981), Simple Approximate Distributional Results for Confidence
and Tolerance Limits for the Weibull Distribution Based on Maximum Likelihood Estimators, Technometrics, 23, 15–20.
Coles, S. (2001), An Introduction to Statistical Modeling of Extreme Values, Springer.
See Also
Weibull
Examples
## 85%/90% 1-sided Weibull tolerance intervals for a sample
## of size 150.
set.seed(100)
x <- rweibull(150, 3, 75)
out <- exttol.int(x = x, alpha = 0.15, P = 0.90, side = 1,
dist = "Weibull")
out
28
F1
plottol(out, x, plot.type = "both", side = "lower",
x.lab = "Weibull Data")
F1
Appell’s F1 Hypergeometric Function
Description
The Appell function of the first kind, which is a two variable extension of the hypergeometric
distribution.
Usage
F1(a, b, b.prime, c, x, y, ...)
Arguments
a, b, b.prime, c
Appropriate parameters for this function.
x, y
The inputted values to evaluate this function such that each is less than 1 in
absolute value.
...
Additional arguments passed to the integrate function.
Value
F1 returns the simple integral result for the Appell function of the first kind with the arguments
specified above.
Note
This function is solved by using a simple integral representation for real numbers. While all four of
the Appell functions can be extended to the complex plane, this is not an option for this code.
References
Bailey, W. N. (1935), Generalised Hypergeometric Series, Cambridge University Press.
See Also
DiffProp, integrate
Examples
## Sample calculation.
F1(a = 3, b = 4, b.prime = 5, c = 13, x = 0.2, y = 0.4)
fidbintol.int
fidbintol.int
29
Fiducial-Based Tolerance Intervals for the Function of Two Binomial
Proportions
Description
Provides 1-sided or 2-sided tolerance intervals for the function of two binomial proportions using
fiducial quantities.
Usage
fidbintol.int(x1, x2, n1, n2, m1 = NULL, m2 = NULL, FUN,
alpha = 0.05, P = 0.99, side = 1, K = 1000,
B = 1000)
Arguments
x1
A value of observed "successes" from group 1.
x2
A value of observed "successes" from group 2.
n1
The total number of trials for group 1.
n2
The total number of trials for group 2.
m1
The total number of future trials for group 1. If NULL, then it is set to n1.
m2
The total number of future trials for group 2. If NULL, then it is set to n2.
FUN
Any reasonable (and meaningful) function taking exactly two arguments that we
are interested in constructing a tolerance interval on.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
K
The number of fiducial quantities to be generated. The number of iterations
should be at least as large as the default value of 1000. See Details for the
definition of the fiducial quantity for a binomial proportion.
B
The number of iterations used for the Monte Carlo algorithm which determines
the tolerance limits. The number of iterations should be at least as large as the
default value of 1000.
Details
If X is observed from a Bin(n, p) distribution, then the fiducial quantity for p is Beta(X +0.5, n−
X + 0.5).
30
fidbintol.int
Value
fidbintol.int returns a list with two items. The first item (tol.limits) is a data frame with the
following items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
fn.est
A point estimate of the functional form of interest using the maximum likelihood
estimates calculated with the inputted values of x1, x2, n1, and n2.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
The second item (fn) simply returns the functional form specified by FUN.
References
Clopper, C. J. and Pearson, E. S. (1934), The Use of Confidence or Fiducial Limits Illustrated in
the Case of the Binomial, Biometrika, 26, 404–413.
Krishnamoorthy, K. and Lee, M. (2010), Inference for Functions of Parameters in Discrete Distributions Based on Fiducial Approach: Binomial and Poisson Cases, Journal of Statistical Planning
and Inference, 140, 1182–1192.
Mathew, T. and Young, D. S. (2013), Fiducial-Based Tolerance Intervals for Some Discrete Distributions, Computational Statistics and Data Analysis, 61, 38–49.
See Also
fidnegbintol.int, fidpoistol.int
Examples
## 95%/99% 1-sided and 2-sided tolerance intervals for
## the difference between binomial proportions.
set.seed(100)
p1 <- 0.2
p2 <- 0.4
n1 <- n2 <- 200
x1 <- rbinom(1, n1, p1)
x2 <- rbinom(1, n2, p2)
fun.ti <- function(x, y) x - y
fidbintol.int(x1, x2,
alpha =
fidbintol.int(x1, x2,
alpha =
n1, n2,
0.05, P
n1, n2,
0.05, P
m1 = 500, m2
= 0.99, side
m1 = 500, m2
= 0.99, side
=
=
=
=
500, FUN = fun.ti,
1)
500, FUN = fun.ti,
2)
fidnegbintol.int
fidnegbintol.int
31
Fiducial-Based Tolerance Intervals for the Function of Two Negative
Binomial Proportions
Description
Provides 1-sided or 2-sided tolerance intervals for the function of two negative binomial proportions
using fiducial quantities.
Usage
fidnegbintol.int(x1, x2, n1, n2, m1 = NULL, m2 = NULL, FUN,
alpha = 0.05, P = 0.99, side = 1, K = 1000,
B = 1000)
Arguments
x1
A value of observed "failures" from group 1.
x2
A value of observed "failures" from group 2.
n1
The target number of successes for group 1.
n2
The target number of successes for group 2.
m1
The total number of future trials for group 1. If NULL, then it is set to n1.
m2
The total number of future trials for group 2. If NULL, then it is set to n2.
FUN
Any reasonable (and meaningful) function taking exactly two arguments that we
are interested in constructing a tolerance interval on.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
K
The number of fiducial quantities to be generated. The number of iterations
should be at least as large as the default value of 1000. See Details for the
definition of the fiducial quantity for a negative binomial proportion.
B
The number of iterations used for the Monte Carlo algorithm which determines
the tolerance limits. The number of iterations should be at least as large as the
default value of 1000.
Details
If X is observed from a N egBin(n, p) distribution, then the fiducial quantity for p is Beta(n, X +
0.5).
32
fidnegbintol.int
Value
fidnegbintol.int returns a list with two items. The first item (tol.limits) is a data frame with
the following items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
fn.est
A point estimate of the functional form of interest using the maximum likelihood
estimates calculated with the inputted values of x1, x2, n1, and n2.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
The second item (fn) simply returns the functional form specified by FUN.
References
Cai, Y. and Krishnamoorthy, K. (2005), A Simple Improved Inferential Method for Some Discrete
Distributions, Computational Statistics and Data Analysis, 48, 605–621.
Clopper, C. J. and Pearson, E. S. (1934), The Use of Confidence or Fiducial Limits Illustrated in
the Case of the Binomial, Biometrika, 26, 404–413.
Krishnamoorthy, K. and Lee, M. (2010), Inference for Functions of Parameters in Discrete Distributions Based on Fiducial Approach: Binomial and Poisson Cases, Journal of Statistical Planning
and Inference, 140, 1182–1192.
Mathew, T. and Young, D. S. (2013), Fiducial-Based Tolerance Intervals for Some Discrete Distributions, Computational Statistics and Data Analysis, 61, 38–49.
See Also
fidbintol.int, fidpoistol.int
Examples
## 95%/99% 1-sided and 2-sided tolerance intervals for
## the ratio of odds ratios for negative binomial proportions.
set.seed(100)
p1 <- 0.6
p2 <- 0.2
n1 <- n2 <- 50
x1 <- rnbinom(1, n1, p1)
x2 <- rnbinom(1, n2, p2)
fun.ti <- function(x, y) x * (1 - y) / (y * (1 - x))
fidnegbintol.int(x1, x2, n1, n2, m1 = 50, m2 = 50, FUN = fun.ti,
alpha = 0.05, P = 0.99, side = 1)
fidnegbintol.int(x1, x2, n1, n2, m1 = 50, m2 = 50, FUN = fun.ti,
fidpoistol.int
33
alpha = 0.05, P = 0.99, side = 2)
fidpoistol.int
Fiducial-Based Tolerance Intervals for the Function of Two Poisson
Rates
Description
Provides 1-sided or 2-sided tolerance intervals for the function of two Poisson rates using fiducial
quantities.
Usage
fidpoistol.int(x1, x2, n1, n2, m1 = NULL, m2 = NULL, FUN,
alpha = 0.05, P = 0.99, side = 1, K = 1000,
B = 1000)
Arguments
x1
A value of observed counts from group 1.
x2
A value of observed counts from group 2.
n1
The length of time that x1 was recorded over.
n2
The length of time that x2 was recorded over.
m1
The total number of future trials for group 1. If NULL, then it is set to n1.
m2
The total number of future trials for group 2. If NULL, then it is set to n2.
FUN
Any reasonable (and meaningful) function taking exactly two arguments that we
are interested in constructing a tolerance interval on.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
K
The number of fiducial quantities to be generated. The number of iterations
should be at least as large as the default value of 1000. See Details for the
definition of the fiducial quantity for a Poisson rate.
B
The number of iterations used for the Monte Carlo algorithm which determines
the tolerance limits. The number of iterations should be at least as large as the
default value of 1000.
Details
If X is observed from a P oi(n ∗ λ) distribution, then the fiducial quantity for λ is χ22∗x+1 /(2 ∗ n).
34
fidpoistol.int
Value
fidpoistol.int returns a list with two items. The first item (tol.limits) is a data frame with the
following items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
fn.est
A point estimate of the functional form of interest using the maximum likelihood
estimates calculated with the inputted values of x1, x2, n1, and n2.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
The second item (fn) simply returns the functional form specified by FUN.
References
Cox, D. R. (1953), Some Simple Approximate Tests for Poisson Variates, Biometrika, 40, 354–360.
Krishnamoorthy, K. and Lee, M. (2010), Inference for Functions of Parameters in Discrete Distributions Based on Fiducial Approach: Binomial and Poisson Cases, Journal of Statistical Planning
and Inference, 140, 1182–1192.
Mathew, T. and Young, D. S. (2013), Fiducial-Based Tolerance Intervals for Some Discrete Distributions, Computational Statistics and Data Analysis, 61, 38–49.
See Also
fidbintol.int, fidnegbintol.int
Examples
## 95%/99% 1-sided and 2-sided tolerance intervals for
## the ratio of two Poisson rates.
set.seed(100)
lambda1 <- 10
lambda2 <- 2
n1 <- 3000
n2 <- 3250
x1 <- rpois(1, n1 * lambda1)
x2 <- rpois(1, n2 * lambda2)
fun.ti <- function(x, y) x / y
fidpoistol.int(x1,
FUN
fidpoistol.int(x1,
FUN
x2, n1, n2, m1 = 2000, m2
= fun.ti, alpha = 0.05, P
x2, n1, n2, m1 = 2000, m2
= fun.ti, alpha = 0.05, P
=
=
=
=
2500,
0.99, side = 1)
2500,
0.99, side = 2)
gamtol.int
gamtol.int
35
Gamma (or Log-Gamma) Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for data distributed according to either a gamma
distribution or log-gamma distribution.
Usage
gamtol.int(x, alpha = 0.05, P = 0.99, side = 1,
method = c("HE", "HE2", "WBE", "ELL", "KM", "EXACT",
"OCT"), m = 50, log.gamma = FALSE)
Arguments
x
A vector of data which is distributed according to either a gamma distribution
or a log-gamma distribution.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
method
The method for calculating the k-factors. The k-factor for the 1-sided tolerance intervals is performed exactly and thus is the same for the chosen method.
"HE" is the Howe method and is often viewed as being extremely accurate, even
for small sample sizes. "HE2" is a second method due to Howe, which performs similarly to the Weissberg-Beatty method, but is computationally simpler. "WBE" is the Weissberg-Beatty method (also called the Wald-Wolfowitz
method), which performs similarly to the first Howe method for larger sample
sizes. "ELL" is the Ellison correction to the Weissberg-Beatty method when f is
appreciably larger than n^2. A warning message is displayed if f is not larger
than n^2. "KM" is the Krishnamoorthy-Mathew approximation to the exact solution, which works well for larger sample sizes. "EXACT" computes the k-factor
exactly by finding the integral solution to the problem via the integrate function. Note the computation time of this method is largely determined by m.
"OCT" is the Owen approach to compute the k-factor when controlling the tails
so that there is not more than (1-P)/2 of the data in each tail of the distribution.
m
The maximum number of subintervals to be used in the integrate function.
This is necessary only for method = "EXACT" and method = "OCT". The larger
the number, the more accurate the solution. Too low of a value can result in an error. A large value can also cause the function to be slow for method = "EXACT".
log.gamma
If TRUE, then the data is considered to be from a log-gamma distribution, in
which case the output gives tolerance intervals for the log-gamma distribution.
The default is FALSE.
36
gamtol.int
Details
Recall that if the random variable X is distributed according to a log-gamma distribution, then the
random variable Y = ln(X) is distributed according to a gamma distribution.
Value
gamtol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Krishnamoorthy, K., Mathew, T., and Mukherjee, S. (2008), Normal-Based Methods for a Gamma
Distribution: Prediction and Tolerance Intervals and Stress-Strength Reliability, Technometrics, 50,
69–78.
See Also
GammaDist, K.factor
Examples
## 99%/99% 1-sided gamma tolerance intervals for a sample
## of size 50.
set.seed(100)
x <- rgamma(50, 0.30, scale = 2)
out <- gamtol.int(x = x, alpha = 0.01, P = 0.99, side = 1,
method = "HE")
out
plottol(out, x, plot.type = "both", side = "upper",
x.lab = "Gamma Data")
hypertol.int
hypertol.int
37
Hypergeometric Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for hypergeometric random variables. From a sampling without replacement perspective, these limits use the proportion of units from group A (e.g.,
"black balls" in an urn) in a sample to bound the number of potential units drawn from group A in
a future sample taken from the universe.
Usage
hypertol.int(x, n, N, m = NULL, alpha = 0.05, P = 0.99,
side = 1, method = c("EX", "LS", "CC"))
Arguments
x
n
N
m
alpha
P
side
method
The number of units from group A in the sample. Can be a vector, in which case
the sum of x is used.
The size of the random sample of units selected.
The population size.
The quantity of units to be sampled from the universe for a future study. If
m = NULL, then the tolerance limits will be constructed assuming n for this
quantity.
The level chosen such that 1-alpha is the confidence level.
The proportion of units from group A in future samples of size m to be covered
by this tolerance interval.
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
The method for calculating the lower and upper confidence bounds, which are
used in the calculation of the tolerance bounds. The default method is "EX",
which is an exact-based method. "LS" is the large-sample method. "CC" gives a
continuity-corrected version of the large-sample method.
Value
hypertol.int returns a data frame with items:
alpha
P
rate
p.hat
1-sided.lower
1-sided.upper
2-sided.lower
2-sided.upper
The specified significance level.
The proportion of units from group A in future samples of size m.
The sampling rate determined by n/N.
The proportion of units in the sample from group A, calculated by x/n.
The 1-sided lower tolerance bound. This is given only if side = 1.
The 1-sided upper tolerance bound. This is given only if side = 1.
The 2-sided lower tolerance bound. This is given only if side = 2.
The 2-sided upper tolerance bound. This is given only if side = 2.
38
K.factor
Note
As this methodology is built using large-sample theory, if the sampling rate is less than 0.05, then
a warning is generated stating that the results are not reliable. Also, compare the functionality of
this procedure with the acc.samp procedure, which is to determine a minimal acceptance limit for
a particular sampling plan.
References
Brown, L. D., Cai, T. T., and DasGupta, A. (2001), Interval Estimation for a Binomial Proportion,
Statistical Science, 16, 101–133.
Eichenberger, P., Hulliger, B., and Potterat, J. (2011), Two Measures for Sample Size Determination, Survey Research Methods, 5, 27–37.
Young, D. S. (2014), Tolerance Intervals for Hypergeometric and Negative Hypergeometric Variables, Sankhya: The Indian Journal of Statistics, Series B, to appear.
See Also
acc.samp, Hypergeometric
Examples
## 90%/95% 1-sided and 2-sided hypergeometric tolerance
## intervals for a future sample of 30 when the universe
## is of size 100.
hypertol.int(x
P
hypertol.int(x
P
hypertol.int(x
P
hypertol.int(x
P
K.factor
=
=
=
=
=
=
=
=
15, n
0.95,
15, n
0.95,
15, n
0.95,
15, n
0.95,
= 50, N =
side = 1,
= 50, N =
side = 1,
= 50, N =
side = 2,
= 50, N =
side = 2,
100, m
method
100, m
method
100, m
method
100, m
method
=
=
=
=
=
=
=
=
30, alpha
"LS")
30, alpha
"CC")
30, alpha
"LS")
30, alpha
"CC")
= 0.10,
= 0.10,
= 0.10,
= 0.10,
Estimating K-factors for Tolerance Intervals Based on Normality
Description
Estimates k-factors for tolerance intervals based on normality.
Usage
K.factor(n, f = NULL, alpha = 0.05, P = 0.99, side = 1,
method = c("HE", "HE2", "WBE", "ELL", "KM", "EXACT",
"OCT"), m = 50)
K.factor
39
Arguments
n
The (effective) sample size.
f
The number of degrees of freedom associated with calculating the estimate of
the population standard deviation. If NULL, then f is taken to be n-1.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by the tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
method
The method for calculating the k-factors. The k-factor for the 1-sided tolerance intervals is performed exactly and thus is the same for the chosen method.
"HE" is the Howe method and is often viewed as being extremely accurate, even
for small sample sizes. "HE2" is a second method due to Howe, which performs similarly to the Weissberg-Beatty method, but is computationally simpler. "WBE" is the Weissberg-Beatty method (also called the Wald-Wolfowitz
method), which performs similarly to the first Howe method for larger sample
sizes. "ELL" is the Ellison correction to the Weissberg-Beatty method when f is
appreciably larger than n^2. A warning message is displayed if f is not larger
than n^2. "KM" is the Krishnamoorthy-Mathew approximation to the exact solution, which works well for larger sample sizes. "EXACT" computes the k-factor
exactly by finding the integral solution to the problem via the integrate function. Note the computation time of this method is largely determined by m.
"OCT" is the Owen approach to compute the k-factor when controlling the tails
so that there is not more than (1-P)/2 of the data in each tail of the distribution.
m
The maximum number of subintervals to be used in the integrate function.
This is necessary only for method = "EXACT" and method = "OCT". The larger
the number, the more accurate the solution. Too low of a value can result in an error. A large value can also cause the function to be slow for method = "EXACT".
Value
K.factor returns the k-factor for tolerance intervals based on normality with the arguments specified above.
Note
For larger sample sizes, there may be some accuracy issues with the 1-sided calculation since it
depends on the noncentral t-distribution. The code is primarily intended to be used for moderate
values of the noncentrality parameter. It will not be highly accurate, especially in the tails, for large
values. See TDist for further details.
References
Ellison, B. E. (1964), On Two-Sided Tolerance Intervals for a Normal Distribution, Annals of Mathematical Statistics, 35, 762–772.
Howe, W. G. (1969), Two-Sided Tolerance Limits for Normal Populations - Some Improvements,
Journal of the American Statistical Association, 64, 610–620.
40
K.table
Krishnamoorthy, K. and Mathew, T. (2009), Statistical Tolerance Regions: Theory, Applications,
and Computation, Wiley.
Odeh, R. E. and Owen, D. B. (1980), Tables for Normal Tolerance Limits, Sampling Plans, and
Screening, Marcel-Dekker.
Owen, D. B. (1964), Controls of Percentages in Both Tails of the Normal Distribution, Technometrics, 6, 377-387.
Wald, A. and Wolfowitz, J. (1946), Tolerance Limits for a Normal Distribution, Annals of the
Mathematical Statistics, 17, 208–215.
Weissberg, A. and Beatty, G. (1969), Tables of Tolerance Limit Factors for Normal Distributions,
Technometrics, 2, 483–500.
See Also
integrate, K.table, normtol.int, TDist
Examples
## Showing the effect of the Howe, Weissberg-Beatty,
## and exact estimation methods as the sample size increases.
K.factor(10, P = 0.95, side = 2, method = "HE")
K.factor(10, P = 0.95, side = 2, method = "WBE")
K.factor(10, P = 0.95, side = 2, method = "EXACT", m = 50)
K.factor(100, P = 0.95, side = 2, method = "HE")
K.factor(100, P = 0.95, side = 2, method = "WBE")
K.factor(100, P = 0.95, side = 2, method = "EXACT", m = 50)
K.table
Tables of K-factors for Tolerance Intervals Based on Normality
Description
Tabulated summary of k-factors for tolerance intervals based on normality. The user can specify
multiple values for each of the three inputs.
Usage
K.table(n, alpha, P, side = 1, f = NULL, method = c("HE",
"HE2", "WBE", "ELL", "KM", "EXACT", "OCT"), m = 50,
by.arg = c("n", "alpha", "P"))
K.table
41
Arguments
n
A vector of (effective) sample sizes.
alpha
The level chosen such that 1-alpha is the confidence level. Can be a vector.
P
The proportion of the population to be covered by this tolerance interval. Can
be a vector.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
f
The number of degrees of freedom associated with calculating the estimate of
the population standard deviation. If NULL, then f is taken to be n-1. Only a
single value can be specified for f.
method
The method for calculating the k-factors. The k-factor for the 1-sided tolerance intervals is performed exactly and thus is the same for the chosen method.
"HE" is the Howe method and is often viewed as being extremely accurate, even
for small sample sizes. "HE2" is a second method due to Howe, which performs similarly to the Weissberg-Beatty method, but is computationally simpler. "WBE" is the Weissberg-Beatty method (also called the Wald-Wolfowitz
method), which performs similarly to the first Howe method for larger sample
sizes. "ELL" is the Ellison correction to the Weissberg-Beatty method when f is
appreciably larger than n^2. A warning message is displayed if f is not larger
than n^2. "KM" is the Krishnamoorthy-Mathew approximation to the exact solution, which works well for larger sample sizes. "EXACT" computes the k-factor
exactly by finding the integral solution to the problem via the integrate function. Note the computation time of this method is largely determined by m.
"OCT" is the Owen approach to compute the k-factor when controlling the tails
so that there is not more than (1-P)/2 of the data in each tail of the distribution.
m
The maximum number of subintervals to be used in the integrate function.
This is necessary only for method = "EXACT" and method = "OCT". The larger
the number, the more accurate the solution. Too low of a value can result in an error. A large value can also cause the function to be slow for method = "EXACT".
by.arg
How you would like the output organized. If by.arg = "n", then the output
provides a list of matrices sorted by the values specified in n. The matrices have
rows corresponding to the values specified by 1-alpha and columns corresponding to the values specified by P. If by.arg = "alpha", then the output provides
a list of matrices sorted by the values specified in 1-alpha. The matrices have
rows corresponding to the values specified by n and columns corresponding to
the values specified by P. If by.arg = "P", then the output provides a list of
matrices sorted by the values specified in P. The matrices have rows corresponding to the values specified by 1-alpha and columns corresponding to the values
specified by n.
Details
The method used for estimating the k-factors is that due to Howe as it is generally viewed as more
accurate than the Weissberg-Beatty method.
42
laptol.int
Value
K.table returns a list with a structure determined by the argument by.arg described above.
References
Howe, W. G. (1969), Two-Sided Tolerance Limits for Normal Populations - Some Improvements,
Journal of the American Statistical Association, 64, 610–620.
Weissberg, A. and Beatty, G. (1969), Tables of Tolerance Limit Factors for Normal Distributions,
Technometrics, 2, 483–500.
See Also
K.factor
Examples
## Tables generated for each value of the sample size.
K.table(n = seq(50, 100, 10), alpha = c(0.01, 0.05, 0.10),
P = c(0.90, 0.95, 0.99), by.arg = "n")
## Tables generated for each value of the confidence level.
K.table(n = seq(50, 100, 10), alpha = c(0.01, 0.05, 0.10),
P = c(0.90, 0.95, 0.99), by.arg = "alpha")
## Tables generated for each value of the coverage proportion.
K.table(n = seq(50, 100, 10), alpha = c(0.01, 0.05, 0.10),
P = c(0.90, 0.95, 0.99), by.arg = "P")
laptol.int
Laplace Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for data distributed according to a Laplace distribution.
Usage
laptol.int(x, alpha = 0.05, P = 0.99, side = 1)
laptol.int
43
Arguments
x
A vector of data which is distributed according to a Laplace distribution.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
Value
laptol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Bain, L. J. and Engelhardt, M. (1973), Interval Estimation for the Two Parameter Double Exponential Distribution, Technometrics, 15, 875–887.
Examples
## First generate data from a Laplace distribution with location
## parameter 70 and scale parameter 3.
set.seed(100)
tmp <- runif(40)
x <- rep(70, 40) - sign(tmp - 0.5)*rep(3, 40)*
log(2*ifelse(tmp < 0.5, tmp, 1-tmp))
## 95%/90% 1-sided Laplace tolerance intervals for the sample
## of size 40 generated above.
out <- laptol.int(x = x, alpha = 0.05, P = 0.90, side = 1)
out
plottol(out, x, plot.type = "hist", side = "two",
x.lab = "Laplace Data")
44
logistol.int
logistol.int
Logistic (or Log-Logistic) Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for data distributed according to a logistic or loglogistic distribution.
Usage
logistol.int(x, alpha = 0.05, P = 0.99, log.log = FALSE,
side = 1)
Arguments
x
A vector of data which is distributed according to a logistic or log-logistic distribution.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
log.log
If TRUE, then the data is considered to be from a log-logistic distribution, in
which case the output gives tolerance intervals for the log-logistic distribution.
The default is FALSE.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
Details
Recall that if the random variable X is distributed according to a log-logistic distribution, then the
random variable Y = ln(X) is distributed according to a logistic distribution.
Value
logistol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Balakrishnan, N. (1992), Handbook of the Logistic Distribution, Marcel Dekker, Inc.
Hall, I. J. (1975), One-Sided Tolerance Limits for a Logistic Distribution Based on Censored Samples, Biometrics, 31, 873–880.
mvregtol.region
45
See Also
Logistic
Examples
## 90%/95% 1-sided logistic tolerance intervals for a sample
## of size 20.
set.seed(100)
x <- rlogis(20, 5, 1)
out <- logistol.int(x = x, alpha = 0.10, P = 0.95,
log.log = FALSE, side = 1)
out
plottol(out, x, plot.type = "control", side = "two",
x.lab = "Logistic Data")
mvregtol.region
Multivariate (Multiple) Linear Regression Tolerance Regions
Description
Determines the appropriate tolerance factor for computing multivariate (multiple) linear regression
tolerance regions based on Monte Carlo simulation.
Usage
mvregtol.region(y, x, new.x = NULL, int = TRUE, alpha = 0.05,
P = 0.99, B = 1000)
Arguments
y
An nxq matrix of responses assumed to be drawn from a q-dimensional multivariate normal distribution. n pertains to the sample size.
x
An nxm matrix of predictors (i.e., the design matrix). m pertains to the number
of predictors. Do NOT include a column of 1’s if assuming an intercept. This is
controlled by the int argument.
new.x
A matrix of new values for which to approximate k-factors. This must be a
matrix with m columns.
int
If int = TRUE, then an intercept is assumed.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance region.
B
The number of iterations used for the Monte Carlo algorithm which determines
the tolerance factor. The number of iterations should be at least as large as the
default value of 1000.
46
mvtol.region
Details
A basic sketch of how the algorithm works is as follows:
(1) Generate independent chi-square random variables and Wishart random matrices.
(2) Compute the eigenvalues of the randomly generated Wishart matrices.
(3) Iterate the above steps to generate a set of B sample values such that the 100(1-alpha)-th
percentile is an approximate tolerance factor.
Value
mvregtol.region returns a matrix where the first column is the k-factor, the next q columns are
the estimated responses from the least squares fit, and the final m columns are the predictor values.
The first n rows of the matrix pertain to the raw data as specified by y and x. If values for new.x are
specified, then there is one additional row appended to this output for each row in the matrix new.x.
References
Anderson, T. W. (2003) An Introduction to Multivariate Statistical Analysis, Third Edition, Wiley.
Krishnamoorthy, K. and Mathew, T. (2009), Statistical Tolerance Regions: Theory, Applications,
and Computation, Wiley.
Krishnamoorthy, K. and Mondal, S. (2008), Tolerance Factors in Multiple and Multivariate Linear
Regressions, Communications in Statistics - Simulation and Computation, 37, 546–559.
Examples
## 95%/95% multivariate regression tolerance factors using
## a fertilizer data set presented in Anderson (2003, p. 374).
grain <- c(40, 17, 9, 15, 6, 12, 5, 9)
straw <- c(53, 19, 10, 29, 13, 27, 19, 30)
fert <- c(24, 11, 5, 12, 7, 14, 11, 18)
Y <- cbind(grain, straw)
X <- cbind(fert)
new.x <- c(10, 15, 20)
set.seed(100)
out <- mvregtol.region(Y, X, new.x, int = TRUE, alpha = 0.05,
P = 0.95, B = 5000)
out
mvtol.region
Multivariate Normal Tolerance Regions
Description
Determines the appropriate tolerance factor for computing multivariate normal tolerance regions
based on Monte Carlo methods or other approximations.
mvtol.region
47
Usage
mvtol.region(x, alpha = 0.05, P = 0.99, B = 1000, M = 1000,
method = c("KM", "AM", "GM", "HM", "MHM", "V11",
"HM.V11", "MC"))
Arguments
x
alpha
P
B
M
method
An nxp matrix of data assumed to be drawn from a p-dimensional multivariate
normal distribution. n pertains to the sample size.
The level chosen such that 1-alpha is the confidence level. A vector of alpha
values may be specified.
The proportion of the population to be covered by this tolerance region. A vector
of P values may be specified.
The number of iterations used for the Monte Carlo algorithms (i.e., when method = "KM"
or "MC"), which determines the tolerance factor. The number of iterations should
be at least as large as the default value of 1000.
The number of iterations used for the inner loop of the Monte Carlo algorithm
specified through method = "MC". The number of iterations should be at
least as large as the default value of 1000. Note that this is not required for
method = "KM" since that algorithm handles the eigenvalues differently in the
estimation of the tolerance factor.
The method for estimating the tolerance factors. "KM" is the KrishnamoorthyMondal method, which is the method implemented in previous versions of the
tolerance package. It is one of the more accurate methods available. "AM" is
an approximation method based on the arithmetic mean. "GM" is an approximation method based on the geometric mean. "HM" is an approximation method
based on the harmonic mean. "MHM" is a modified approach based on the harmonic mean. "V11" is a method that utilizes a certain partitioning of a Wishart
random matrix for deriving an approximate tolerance factor. "HM.V11" is a hybrid method of the "HM" and "V11" methods. "MC" is a simple Monte Carlo
approach to estimating the tolerance factor, which is computationally expensive
as the values of B and M increase.
Details
All of the methods are outlined in the references that we provided. In practice, we recommend
using the Krishnamoorthy-Mondal approach. A basic sketch of how the Krishnamoorthy-Mondal
algorithm works is as follows:
(1) Generate independent chi-square random variables and Wishart random matrices.
(2) Compute the eigenvalues of the randomly generated Wishart matrices.
(3) Iterate the above steps to generate a set of B sample values such that the 100(1-alpha)-th
percentile is an approximate tolerance factor.
Value
mvtol.region returns a matrix where the rows pertain to each confidence level 1-alpha specified
and the columns pertain to each proportion level P specified.
48
negbintol.int
References
Krishnamoorthy, K. and Mathew, T. (1999), Comparison of Approximation Methods for Computing
Tolerance Factors for a Multivariate Normal Population, Technometrics, 41, 234–249.
Krishnamoorthy, K. and Mondal, S. (2006), Improved Tolerance Factors for Multivariate Normal
Distributions, Communications in Statistics - Simulation and Computation, 35, 461–478.
Examples
## 90%/90% bivariate normal tolerance region.
set.seed(100)
x1 <- rnorm(100, 0, 0.2)
x2 <- rnorm(100, 0, 0.5)
x <- cbind(x1, x2)
out1 <- mvtol.region(x = x, alpha = 0.10, P = 0.90, B = 1000,
method = "KM")
out1
plottol(out1, x)
## 90%/90% trivariate normal tolerance region.
set.seed(100)
x1 <- rnorm(100, 0, 0.2)
x2 <- rnorm(100, 0, 0.5)
x3 <- rnorm(100, 5, 1)
x <- cbind(x1, x2, x3)
mvtol.region(x = x, alpha = c(0.10, 0.05, 0.01),
P = c(0.90, 0.95, 0.99), B = 1000, method = "KM")
out2 <- mvtol.region(x = x, alpha = 0.10, P = 0.90, B = 1000,
method = "KM")
out2
plottol(out2, x)
negbintol.int
Negative Binomial Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for negative binomial random variables. From a
statistical quality control perspective, these limits use the number of failures that occur to reach n
successes to bound the number of failures for a specified amount of future successes (m).
Usage
negbintol.int(x, n, m = NULL, alpha = 0.05, P = 0.99,
side = 1, method = c("LS", "WU", "CB",
"CS", "SC", "LR", "SP", "CC"))
negbintol.int
49
Arguments
x
The total number of failures that occur from a sample of size n. Can be a vector
of length n, in which case the sum of x is computed.
n
The target number of successes (sometimes called size) for each trial.
m
The target number of successes in a future lot for which the tolerance limits
will be calculated. If m = NULL, then the tolerance limits will be constructed
assuming n for the target number of future successes.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the defective (or acceptable) units in future samples of size m
to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
method
The method for calculating the lower and upper confidence bounds, which are
used in the calculation of the tolerance bounds. The default method is "LS",
which is the large-sample method based on the MLE. "WU" is a Wald-type interval based on the UMVUE of the negative binomial proportion. "CB" is the
Casella-Berger exact method. "CS" is a method based on chi-square percentiles.
"SC" is the score method. "LR" is a likelihood ratio-based method. "SP" is a
method using a saddlepoint approximation for the confidence intervals. "CC"
gives a continuity-corrected version of the large-sample method and is appropriate when n is large. More information on these methods can be found in the
"References".
Details
This function takes the approach for Poisson and binomial random variables developed in Hahn and
Chandra (1981) and applies it to the negative binomial case.
Value
negbintol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of defective (or acceptable) units in future samples of size m.
pi.hat
The probability of success in each trial, calculated by n/(n+x).
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
Note
Recall that the geometric distribution is the negative binomial distribution where the size is 1. Therefore, the case when n = m = 1 will provide tolerance limits for a geometric distribution.
50
NegHypergeometric
References
Casella, G. and Berger, R. L. (1990), Statistical Inference, Duxbury Press.
Hahn, G. J. and Chandra, R. (1981), Tolerance Intervals for Poisson and Binomial Variables, Journal of Quality Technology, 13, 100–110.
Tian, M., Tang, M. L., Ng, H. K. T., and Chan, P. S. (2009), A Comparative Study of Confidence
Intervals for Negative Binomial Proportions, Journal of Statistical Computation and Simulation,
79, 241–249.
Young, D. S. (2014), A Procedure for Approximate Negative Binomial Tolerance Intervals, Journal
of Statistical Computation and Simulation, 84, 438–450.
See Also
NegBinomial, umatol.int
Examples
## Comparison of 95%/99% 1-sided tolerance limits with
## 50 failures before 10 successes are reached.
negbintol.int(x
negbintol.int(x
negbintol.int(x
negbintol.int(x
negbintol.int(x
negbintol.int(x
negbintol.int(x
negbintol.int(x
##
##
##
##
=
=
=
=
=
=
=
=
50,
50,
50,
50,
50,
50,
50,
50,
n
n
n
n
n
n
n
n
=
=
=
=
=
=
=
=
10,
10,
10,
10,
10,
10,
10,
10,
side
side
side
side
side
side
side
side
=
=
=
=
=
=
=
=
1,
1,
1,
1,
1,
1,
1,
1,
method
method
method
method
method
method
method
method
=
=
=
=
=
=
=
=
"LS")
"WU")
"CB")
"CS")
"SC")
"LR")
"SP")
"CC")
95%/99% 1-sided tolerance limits and 2-sided tolerance
interval for the same setting above, but when we are
interested in a future experiment that requires 20 successes
be reached for each trial.
negbintol.int(x = 50, n = 10, m = 20, side = 1)
negbintol.int(x = 50, n = 10, m = 20, side = 2)
NegHypergeometric
The Negative Hypergeometric Distribution
Description
Density, distribution function, quantile function, and random generation for the negative hypergeometric distribution.
NegHypergeometric
51
Usage
dnhyper(x, m, n, k, log = FALSE)
pnhyper(q, m, n, k, lower.tail = TRUE, log.p = FALSE)
qnhyper(p, m, n, k, lower.tail = TRUE, log.p = FALSE)
rnhyper(nn, m, n, k)
Arguments
x,q
Vector of quantiles representing the number of trials until k successes have occurred (e.g., until k white balls have been drawn from an urn without replacement).
m
The number of successes in the population (e.g., the number of white balls in
the urn).
n
The population size (e.g., the total number of balls in the urn).
k
The number of successes (e.g., white balls) to achieve with the sample.
p
Vector of probabilities, which must be between 0 and 1.
nn
The number of observations. If length>1, then the length is taken to be the
number required.
log,log.p
Logical vectors. If TRUE, then probabilities are given as log(p).
lower.tail
Logical vector. If TRUE, then probabilities are P [X ≤ x], else P [X > x].
Details
A negative hypergeometric distribution (sometimes called the inverse hypergeometric distribution)
models the total number of trials until k successes occur. Compare this to the negative binomial
distribution, which models the number of failures that occur until a specified number of successes
has been reached. The negative hypergeometric distribution has density
x−1 n−x
p(x) =
k−1
m−k
n
m
for x = k, k + 1, ..., n − m + k.
Value
dnhyper gives the density, pnhyper gives the distribution function, qnhyper gives the quantile
function, and rnhyper generates random deviates.
Invalid arguments will return value NaN, with a warning.
References
Wilks, S. S. (1963), Mathematical Statistics, Wiley.
See Also
runif and .Random.seed about random number generation.
52
neghypertol.int
Examples
## Randomly generated data from the negative hypergeometric
## distribution.
set.seed(100)
x <- rnhyper(nn = 1000, m = 15, n = 40, k = 10)
hist(x, main = "Randomly Generated Data", prob = TRUE)
x.1 = sort(x)
y <- dnhyper(x = x.1, m = 15, n = 40, k = 10)
lines(x.1, y, col = 2, lwd = 2)
plot(x.1, pnhyper(q = x.1, m = 15, n = 40, k = 10),
type = "l", xlab = "x", ylab = "Cumulative Probabilities")
qnhyper(p = 0.20, m = 15, n = 40, k = 10, lower.tail = FALSE)
qnhyper(p = 0.80, m = 15, n = 40, k = 10)
neghypertol.int
Negative Hypergeometric Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for negative hypergeometric random variables. When
sampling without replacement, these limits are on the total number of expected draws in a future
sample in order to achieve a certain number from group A (e.g., "black balls" in an urn).
Usage
neghypertol.int(x, n, N, m = NULL, alpha = 0.05, P = 0.99,
side = 1, method = c("EX", "LS", "CC"))
Arguments
x
The number of units drawn in order to achieve n successes. Can be a vector, in
which case the sum of x is used.
n
The target number of successes in the sample drawn (e.g., the number of "black
balls" you are to draw in the sample).
N
The population size (e.g., the total number of balls in the urn).
m
The target number of successes to be sampled from the universe for a future
study. If m = NULL, then the tolerance limits will be constructed assuming n for
this quantity.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of units from group A in future samples of size m to be covered
by this tolerance interval.
neghypertol.int
53
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
method
The method for calculating the lower and upper confidence bounds, which are
used in the calculation of the tolerance bounds. The default method is "EX",
which is an exact-based method. "LS" is the large-sample method. "CC" gives a
continuity-corrected version of the large-sample method.
Value
neghypertol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of units from group A in future samples of size m.
rate
The sampling rate determined by x/N.
p.hat
The proportion of units in the sample from group A, calculated by n/x.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
Note
As this methodology is built using large-sample theory, if the sampling rate is less than 0.05, then a
warning is generated stating that the results are not reliable.
References
Khan, R. A. (1994), A Note on the Generating Function of a Negative Hypergeometric Distribution,
Sankhya: The Indian Journal of Statistics, Series B, 56, 309–313.
Young, D. S. (2014), Tolerance Intervals for Hypergeometric and Negative Hypergeometric Variables, Sankhya: The Indian Journal of Statistics, Series B, to appear.
See Also
acc.samp, NegHypergeometric
Examples
##
##
##
##
##
90%/95% 1-sided and 2-sided negative hypergeometric
tolerance intervals for a future number of 300 successes
when the universe is of size 1000. The estimates are
based on having drawn 425 in another sample to achieve
200 successes.
neghypertol.int(425, 200,
P = 0.95,
neghypertol.int(425, 200,
P = 0.95,
1000, m =
side = 1,
1000, m =
side = 1,
300, alpha = 0.05,
method = "LS")
300, alpha = 0.05,
method = "CC")
54
nlregtol.int
neghypertol.int(425, 200,
P = 0.95,
neghypertol.int(425, 200,
P = 0.95,
nlregtol.int
1000, m =
side = 2,
1000, m =
side = 2,
300, alpha = 0.05,
method = "LS")
300, alpha = 0.05,
method = "CC")
Nonlinear Regression Tolerance Bounds
Description
Provides 1-sided or 2-sided nonlinear regression tolerance bounds.
Usage
nlregtol.int(formula, xy.data = data.frame(), x.new = NULL,
side = 1, alpha = 0.05, P = 0.99, maxiter = 50,
...)
Arguments
formula
A nonlinear model formula including variables and parameters.
xy.data
A data frame in which to evaluate the formulas in formula. The first column of
xy.data must be the response variable.
x.new
Any new levels of the predictor(s) for which to report the tolerance bounds. The
number of columns must be 1 less than the number of columns for xy.data.
side
Whether a 1-sided or 2-sided tolerance bound is required (determined by side = 1
or side = 2, respectively).
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by the tolerance bound(s).
maxiter
A positive integer specifying the maximum number of iterations that the nonlinear least squares routine (nls) should run.
...
Optional arguments passed to nls when estimating the nonlinear regression
equation.
Details
It is highly recommended that the user specify starting values for the nls routine.
Value
nlregtol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by the tolerance bound(s).
norm.OC
55
y.hat
The predicted value of the response for the fitted nonlinear regression model.
y
The value of the response given in the first column of xy.data. This data frame
is sorted by this value.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Wallis, W. A. (1951), Tolerance Intervals for Linear Regression, in Second Berkeley Symposium on
Mathematical Statistics and Probability, ed. J. Neyman, Berkeley: University of CA Press, 43–51.
Young, D. S. (2013), Regression Tolerance Intervals, Communications in Statistics - Simulation and
Computation, 42, 2040–2055.
See Also
nls
Examples
## 95%/95% 2-sided nonlinear regression tolerance bounds
## for a sample of size 50.
set.seed(100)
x <- runif(50, 5, 45)
f1 <- function(x, b1, b2) b1 + (0.49 - b1)*exp(-b2*(x - 8)) +
rnorm(50, sd = 0.01)
y <- f1(x, 0.39, 0.11)
formula <- as.formula(y ~ b1 + (0.49 - b1)*exp(-b2*(x - 8)))
out <- nlregtol.int(formula = formula,
xy.data = data.frame(cbind(y, x)),
x.new=cbind(c(10, 20)), side = 2,
alpha = 0.05, P = 0.95)
out
plottol(out, x = x, y = y, side = "two", x.lab = "X",
y.lab = "Y")
norm.OC
Operating Characteristic (OC) Curves for K-Factors for Tolerance Intervals Based on Normality
Description
Provides OC-type curves to illustrate how values of the k-factors for normal tolerance intervals,
confidence levels, and content levels change as a function of the sample size.
56
norm.OC
Usage
norm.OC(k = NULL, alpha = NULL, P = NULL, n, side = 1,
method = c("HE", "HE2", "WBE", "ELL", "KM", "EXACT",
"OCT"), m = 50)
Arguments
k
If wanting OC curves where the confidence level or content level is on the yaxis, then a single positive value of k must be specified. This would be the target
k-factor for the desired tolerance interval. If k = NULL, then OC curves will be
constructed where the k-factor value is found for given levels of alpha, P, and
n.
alpha
The set of levels chosen such that 1-alpha are confidence levels. If wanting
OC curves where the content level is being calculated, then each curve will
correspond to a level in the set of alpha. If a set of P values is specified, then
OC curves will be constructed where the k-factor is found and each curve will
correspond to each combination of alpha and P. If alpha = NULL, then OC
curves will be constructed to find the confidence level for given levels of k, P,
and n.
P
The set of content levels to be considered. If wanting OC curves where the
confidence level is being calculated, then each curve will correspond to a level
in the set of P. If a set of alpha values is specified, then OC curves will be
constructed where the k-factor is found and each curve will correspond to each
combination of alpha and P. If P = NULL, then OC curves will be constructed
to find the content level for given levels of k, alpha, and n.
n
A sequence of sample sizes to consider. This must be a vector of at least length
2 since all OC curves are constructed as functions of n.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
method
The method for calculating the k-factors. The k-factor for the 1-sided tolerance intervals is performed exactly and thus is the same for the chosen method.
"HE" is the Howe method and is often viewed as being extremely accurate, even
for small sample sizes. "HE2" is a second method due to Howe, which performs similarly to the Weissberg-Beatty method, but is computationally simpler. "WBE" is the Weissberg-Beatty method (also called the Wald-Wolfowitz
method), which performs similarly to the first Howe method for larger sample
sizes. "ELL" is the Ellison correction to the Weissberg-Beatty method when f is
appreciably larger than n^2. A warning message is displayed if f is not larger
than n^2. "KM" is the Krishnamoorthy-Mathew approximation to the exact solution, which works well for larger sample sizes. "EXACT" computes the k-factor
exactly by finding the integral solution to the problem via the integrate function. Note the computation time of this method is largely determined by m.
"OCT" is the Owen approach to compute the k-factor when controlling the tails
so that there is not more than (1-P)/2 of the data in each tail of the distribution.
m
The maximum number of subintervals to be used in the integrate function,
which is used for the underlying exact method for calculating the normal tolerance intervals.
norm.ss
57
Value
norm.OC returns a figure with the OC curves constructed using the specifications in the arguments.
References
Young, D. S. (2016), Normal Tolerance Interval Procedures in the tolerance Package, submitted.
See Also
K.factor, normtol.int
Examples
## The three types of OC-curves that can be constructed
## with the norm.OC function.
norm.OC(k = 4, alpha = NULL, P = c(0.90, 0.95, 0.99),
n = 10:20, side = 1)
norm.OC(k = 4, alpha = c(0.01, 0.05, 0.10), P = NULL,
n = 10:20, side = 1)
norm.OC(k = NULL, P = c(0.90, 0.95, 0.99),
alpha=c(0.01,0.05,0.10), n = 10:20, side = 1)
norm.ss
Sample Size Determination for Normal Tolerance Intervals
Description
Provides minimum sample sizes for a future sample size when constructing normal tolerance intervals. Various strategies are available for determining the sample size, including strategies that
incorporate known specification limits.
Usage
norm.ss(x = NULL, alpha = 0.05, P = 0.99, delta = NULL,
P.prime = NULL, side = 1, m = 50, spec = c(NA, NA),
hyper.par = list(mu.0 = NULL, sig2.0 = NULL,
m.0 = NULL, n.0 = NULL), method = c("DIR",
"FW", "YGZO"))
58
norm.ss
Arguments
x
A vector of current data that is distributed according to a normal distribution.
This is only required for method = "YGZO".
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
delta
The precision measure for the future tolerance interval as specified under the
Faulkenberry-Weeks method.
P.prime
The proportion of the population (greater than P) such that the tolerance interval
of interest will only exceed P.prime by the probability given by delta.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
m
The maximum number of subintervals to be used in the integrate function,
which is used for the underlying exact method for calculating the normal tolerance intervals.
spec
A vector of length 2 given known specification limits. These are required when
method = "DIR" or method = "YGZO". By default, the values are NA. The two
elements of the vector are for the lower and upper specification limits, respectively. If side = 1, then only one of the specification limits must be specified.
If side = 2, then both specification limits must be specified.
hyper.par
Necessary parameter values for the different methods. If method = "DIR" or
method = "YGZO", then mu.0 and sig2.0 must be specified, which correspond
to the assumed population mean and variance of the underlying normal distribution, which further pertains to the historical data for method = "YGZO". If
method = "YGZO" and the sample size is to be determined using Bayesian normal tolerance intervals, then this is a required list consisting of the hyperparameters for the conjugate prior – the hyperparameters for the mean (mu.0 and n.0)
and the hyperparameters for the variance (sig2.0 and m.0).
method
The method for performing the sample size determination. "DIR" is the direct method (intended as a simple calculation for planning purposes) where the
mean and standard deviation are taken as truth and the sample size is determined with respect to the given specification limits. "FW" is for the traditional
Faulkenberry-Weeks approach for sample size determination. "YGZO" is for
the Young-Gordon-Zhu-Olin approach, which incorporates historical data and
specification limits for determining the value of delta and/or P.prime in the
Faulkenberry-Weeks approach. Note that for "YGZO", at least one of delta and
P.prime must be NULL.
Value
norm.ss returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
delta
The user-specified or calculated precision measure. Not returned if method = "DIR".
P.prime
The user-specified or calculated closeness measure. Not returned if method = "DIR".
normtol.int
n
59
The minimum sample size determined using the conditions specified for this
function.
References
Faulkenberry, G. D. and Weeks, D. L. (1968), Sample Size Determination for Tolerance Limits,
Technometrics, 10, 343–348.
Young, D. S., Gordon, C. M., Zhu, S., and Olin, B. D. (2016), Sample Size Determination Strategies
for Normal Tolerance Intervals Using Historical Data, Quality Engineering (to appear).
See Also
bayesnormtol.int, Normal, normtol.int
Examples
## Sample size determination for 95%/95% 2-sided normal
## tolerance intervals using the direct method.
set.seed(100)
norm.ss(alpha = 0.05, P = 0.95, side = 2, spec = c(-3, 3),
method = "DIR", hyper.par = list(mu.0 = 0,
sig2.0 = 1))
normtol.int
Normal (or Log-Normal) Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for data distributed according to either a normal
distribution or log-normal distribution.
Usage
normtol.int(x, alpha = 0.05, P = 0.99, side = 1,
method = c("HE", "HE2", "WBE", "ELL", "KM",
"EXACT", "OCT"), m = 50, log.norm = FALSE)
Arguments
x
A vector of data which is distributed according to either a normal distribution or
a log-normal distribution.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
60
normtol.int
method
m
log.norm
The method for calculating the k-factors. The k-factor for the 1-sided tolerance intervals is performed exactly and thus is the same for the chosen method.
"HE" is the Howe method and is often viewed as being extremely accurate, even
for small sample sizes. "HE2" is a second method due to Howe, which performs similarly to the Weissberg-Beatty method, but is computationally simpler. "WBE" is the Weissberg-Beatty method (also called the Wald-Wolfowitz
method), which performs similarly to the first Howe method for larger sample
sizes. "ELL" is the Ellison correction to the Weissberg-Beatty method when f is
appreciably larger than n^2. A warning message is displayed if f is not larger
than n^2. "KM" is the Krishnamoorthy-Mathew approximation to the exact solution, which works well for larger sample sizes. "EXACT" computes the k-factor
exactly by finding the integral solution to the problem via the integrate function. Note the computation time of this method is largely determined by m.
"OCT" is the Owen approach to compute the k-factor when controlling the tails
so that there is not more than (1-P)/2 of the data in each tail of the distribution.
The maximum number of subintervals to be used in the integrate function.
This is necessary only for method = "EXACT" and method = "OCT". The larger
the number, the more accurate the solution. Too low of a value can result in an error. A large value can also cause the function to be slow for method = "EXACT".
If TRUE, then the data is considered to be from a log-normal distribution, in
which case the output gives tolerance intervals for the log-normal distribution.
The default is FALSE.
Details
Recall that if the random variable X is distributed according to a log-normal distribution, then the
random variable Y = ln(X) is distributed according to a normal distribution.
Value
normtol.int returns a data frame with items:
alpha
P
x.bar
1-sided.lower
1-sided.upper
2-sided.lower
2-sided.upper
The specified significance level.
The proportion of the population covered by this tolerance interval.
The sample mean.
The 1-sided lower tolerance bound. This is given only if side = 1.
The 1-sided upper tolerance bound. This is given only if side = 1.
The 2-sided lower tolerance bound. This is given only if side = 2.
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Howe, W. G. (1969), Two-Sided Tolerance Limits for Normal Populations - Some Improvements,
Journal of the American Statistical Association, 64, 610–620.
Wald, A. and Wolfowitz, J. (1946), Tolerance Limits for a Normal Distribution, Annals of Mathematical Statistics, 17, 208–215.
Weissberg, A. and Beatty, G. (1969), Tables of Tolerance Limit Factors for Normal Distributions,
Technometrics, 2, 483–500.
np.order
61
See Also
Normal, K.factor
Examples
## 95%/95% 2-sided normal tolerance intervals for a sample
## of size 100.
set.seed(100)
x <- rnorm(100, 0, 0.2)
out <- normtol.int(x = x, alpha = 0.05, P = 0.95, side = 2,
method = "HE", log.norm = FALSE)
out
plottol(out, x, plot.type = "both", side = "two",
x.lab = "Normal Data")
np.order
Sample Size Determination for Tolerance Limits Based on Order
Statistics
Description
For given values of m, alpha, and P, this function solves the necessary sample size such that the r-th
(or (n-s+1)-th) order statistic is the [100(1-alpha)%, 100(P)%] lower (or upper) tolerance limit
(see the Details section below for further explanation). This function can also report all combinations of order statistics for 2-sided intervals.
Usage
np.order(m, alpha = 0.05, P = 0.99, indices = FALSE)
Arguments
m
See the Details section below for how m is defined.
alpha
1 minus the confidence level attained when it is desired to cover a proportion P
of the population with the order statistics.
P
The proportion of the population to be covered with confidence 1-alpha with
the order statistics.
indices
An optional argument to report all combinations of order statistics indices for
the upper and lower limits of the 2-sided intervals. Note that this can only be
calculated when m>1.
62
npregtol.int
Details
For the 1-sided tolerance limits, m=s+r such that the probability is at least 1-alpha that at least the
proportion P of the population is below the (n-s+1)-th order statistic for the upper limit or above
the r-th order statistic for the lower limit. This means for the 1-sided upper limit that r=1, while for
the 1-sided lower limit it means that s=1. For the 2-sided tolerance intervals, m=s+r such that the
probability is at least 1-alpha that at least the proportion P of the population is between the r-th and
(n-s+1)-th order statistics. Thus, all combinations of r>0 and s>0 such that m=s+r are considered.
Value
If indices = FALSE, then a single number is returned for the necessary sample size such that the
r-th (or (n-s+1)-th) order statistic is the [100(1-alpha)%, 100(P)%] lower (or upper) tolerance
limit. If indices = TRUE, then a list is returned with a single number for the necessary sample size
and a matrix with 2 columns where each row gives the pairs of indices for the order statistics for all
permissible [100(1-alpha)%, 100(P)%] 2-sided tolerance intervals.
References
Hanson, D. L. and Owen, D. B. (1963), Distribution-Free Tolerance Limits Elimination of the
Requirement That Cumulative Distribution Functions Be Continuous, Technometrics, 5, 518–522.
Scheffe, H. and Tukey, J. W. (1945), Non-Parametric Estimation I. Validation of Order Statistics,
Annals of Mathematical Statistics, 16, 187–192.
See Also
nptol.int
Examples
## Only requesting the sample size.
np.order(m = 5, alpha = 0.05, P = 0.95)
## Requesting the order statistics indices as well.
np.order(m = 5, alpha = 0.05, P = 0.95, indices = TRUE)
npregtol.int
Nonparametric Regression Tolerance Bounds
Description
Provides 1-sided or 2-sided nonparametric regression tolerance bounds.
npregtol.int
63
Usage
npregtol.int(x, y, y.hat, side = 1, alpha = 0.05, P = 0.99,
method = c("WILKS", "WALD", "HM"), upper = NULL,
lower = NULL)
Arguments
x
A vector of values for the predictor variable. Currently, this function is only
capable of handling a single predictor.
y
A vector of values for the response variable.
y.hat
A vector of fitted values extracted from a nonparametric smoothing routine.
side
Whether a 1-sided or 2-sided tolerance bound is required (determined by side = 1
or side = 2, respectively).
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by the tolerance bound(s).
method
The method for determining which indices of the ordered residuals will be used
for the tolerance bounds. "WILKS", "WALD", and "HM" are each described in
nptol.int. However, since only one tolerance bound can actually be reported
for this procedure, only the first tolerance bound will be returned. Note that this
is not an issue when method = "WILKS" is used as it only produces one set of
tolerance bounds.
upper
The upper bound of the data. When NULL, then the maximum of x is used.
lower
The lower bound of the data. When NULL, then the minimum of x is used.
Value
npregtol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by the tolerance bound(s).
x
The values of the predictor variable.
y
The values of the response variable.
y.hat
The predicted value of the response for the fitted nonparametric smoothing routine.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Young, D. S. (2013), Regression Tolerance Intervals, Communications in Statistics - Simulation and
Computation, 42, 2040–2055.
64
nptol.int
See Also
loess, nptol.int, spline
Examples
## 95%/95% 2-sided nonparametric regression tolerance bounds
## for a sample of size 50.
set.seed(100)
x <- runif(50, 5, 45)
f1 <- function(x, b1, b2) b1 + (0.49 - b1)*exp(-b2*(x - 8)) +
rnorm(50, sd = 0.01)
y <- f1(x, 0.39, 0.11)
y.hat <- loess(y~x)$fit
out <- npregtol.int(x = x, y = y, y.hat = y.hat, side = 2,
alpha = 0.05, P = 0.95, method = "WILKS")
out
plottol(out, x = x, y = y, y.hat = y.hat, side = "two",
x.lab = "X", y.lab = "Y")
nptol.int
Nonparametric Tolerance Intervals
Description
Provides 1-sided or 2-sided nonparametric (i.e., distribution-free) tolerance intervals for any continuous data set.
Usage
nptol.int(x, alpha = 0.05, P = 0.99, side = 1,
method = c("WILKS", "WALD", "HM", "YM"),
upper = NULL, lower = NULL)
Arguments
x
A vector of data which no distributional assumptions are made. The data is only
assumed to come from a continuous distribution.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
nptol.int
65
method
The method for determining which indices of the ordered observations will be
used for the tolerance intervals. "WILKS" is the Wilks method, which produces
tolerance bounds symmetric about the observed center of the data by using the
beta distribution. "WALD" is the Wald method, which produces (possibly) multiple tolerance bounds for side = 2 (each having at least the specified confidence
level), but is the same as method = "WILKS" for side = 1. "HM" is the HahnMeeker method, which is based on the binomial distribution, but the upper and
lower bounds may exceed the minimum and maximum of the sample data. For
side = 2, this method will yield two intervals if an odd number of observations
are to be trimmed from each side. "YM" is the Young-Mathew method for performing interpolation or extrapolation based on the order statistics. See below
for more information on this method.
upper
The upper bound of the data. When NULL, then the maximum of x is used. If
method = "YM" and extrapolation is performed, then upper will be greater than
the maximum.
lower
The lower bound of the data. When NULL, then the minimum of x is used. If
method = "YM" and extrapolation is performed, then lower will be less than the
maximum.
Details
For the Young-Mathew method, interpolation or extrapolation is performed. When side = 1, two
intervals are given: one based on linear interpolation/extrapolation of order statistics (OS-Based)
and one based on fractional order statistics (FOS-Based). When side = 2, only an interval based
on linear interpolation/extrapolation of order statistics is given.
Value
nptol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Bury, K. (1999), Statistical Distributions in Engineering, Cambridge University Press.
Hahn, G. J. and Meeker, W. Q. (1991), Statistical Intervals: A Guide for Practitioners, WileyInterscience.
Wald, A. (1943), An Extension of Wilks’ Method for Setting Tolerance Limits, The Annals of
Mathematical Statistics, 14, 45–55.
Wilks, S. S. (1941), Determination of Sample Sizes for Setting Tolerance Limits, The Annals of
Mathematical Statistics, 12, 91–96.
66
paretotol.int
Young, D. S. and Mathew, T. (2014), Improved Nonparametric Tolerance Intervals Based on Interpolated and Extrapolated Order Statistics, Journal of Nonparametric Statistics, 26, 415–432.
See Also
distfree.est, npregtol.int
Examples
## 90%/95% 2-sided nonparametric tolerance intervals for a
## sample of size 20.
set.seed(100)
x <- rlogis(20, 5, 1)
out <- nptol.int(x = x, alpha = 0.10, P = 0.95, side = 1,
method = "WILKS", upper = NULL, lower = NULL)
out
plottol(out, x, plot.type = "both", side = "two", x.lab = "X")
paretotol.int
Pareto (or Power Distribution) Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for data distributed according to either a Pareto
distribution or a power distribution (i.e., the inverse Pareto distribution).
Usage
paretotol.int(x, alpha = 0.05, P = 0.99, side = 1,
method = c("GPU", "DUN"), power.dist = FALSE)
Arguments
x
A vector of data which is distributed according to either a Pareto distribution or
a power distribution.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
method
The method for how the upper tolerance bound is approximated when transforming to utilize the relationship with the 2-parameter exponential distribution.
"GPU" is the Guenther-Patil-Upppuluri method. "DUN" is the Dunsmore method,
which was empirically shown to be an improvement for samples greater than or
equal to 8. More information on these methods can be found in the "References".
paretotol.int
power.dist
67
If TRUE, then the data is considered to be from a power distribution, in which
case the output gives tolerance intervals for the power distribution. The default
is FALSE.
Details
Recall that if the random variable X is distributed according to a Pareto distribution, then the random variable Y = ln(X) is distributed according to a 2-parameter exponential distribution. Moreover, if the random variable W is distributed according to a power distribution, then the random
variable X = 1/W is distributed according to a Pareto distribution, which in turn means that the
random variable Y = ln(1/W ) is distributed according to a 2-parameter exponential distribution.
Value
paretotol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Dunsmore, I. R. (1978), Some Approximations for Tolerance Factors for the Two Parameter Exponential Distribution, Technometrics, 20, 317–318.
Engelhardt, M. and Bain, L. J. (1978), Tolerance Limits and Confidence Limits on Reliability for
the Two-Parameter Exponential Distribution, Technometrics, 20, 37–39.
Guenther, W. C., Patil, S. A., and Uppuluri, V. R. R. (1976), One-Sided β-Content Tolerance Factors
for the Two Parameter Exponential Distribution, Technometrics, 18, 333–340.
Krishnamoorthy, K., Mathew, T., and Mukherjee, S. (2008), Normal-Based Methods for a Gamma
Distribution: Prediction and Tolerance Intervals and Stress-Strength Reliability, Technometrics, 50,
69–78.
See Also
TwoParExponential, exp2tol.int
Examples
## 95%/99% 2-sided Pareto tolerance intervals
## for a sample of size 500.
set.seed(100)
x <- exp(r2exp(500, rate = 0.15, shift = 2))
out <- paretotol.int(x = x, alpha = 0.05, P = 0.99, side = 2,
68
plottol
out
method = "DUN", power.dist = FALSE)
plottol(out, x, plot.type = "both", side = "two",
x.lab = "Pareto Data")
plottol
Plotting Capabilities for Tolerance Intervals
Description
Provides control charts and/or histograms for tolerance bounds on continuous data as well as tolerance ellipses for data distributed according to bivariate and trivariate normal distributions. Scatterplots with regression tolerance bounds and interval plots for ANOVA tolerance intervals may also
be produced.
Usage
plottol(tol.out, x, y = NULL, y.hat = NULL,
side = c("two", "upper", "lower"),
plot.type = c("control", "hist", "both"),
x.lab = NULL, y.lab = NULL, z.lab = NULL, ...)
Arguments
tol.out
Output from any continuous (including ANOVA) tolerance interval procedure
or from a regression tolerance bound procedure.
x
Either data from a continuous distribution or the predictors for a regression
model. If this is a design matrix for a linear regression model, then it must be
in matrix form AND include a column of 1’s if there is to be an intercept. Note
that multiple predictors are only allowed if considering polynomial regression.
If the output for tol.out concerns ANOVA tolerance intervals, then x must be
a data frame.
y
The response vector for a regression setting. Leave as NULL if not doing regression tolerance bounds.
y.hat
The fitted values from a nonparametric smoothing routine if plotting nonparametric regression tolerance bounds. Otherwise, leave as NULL.
side
side = "two" produces plots for either the two-sided tolerance intervals or
both one-sided tolerance intervals. This will be determined by the output in
tol.out. side = "upper" produces plots showing the upper tolerance bounds.
side = "lower" produces plots showing the lower tolerance bounds.
plot.type
plot.type = "control" produces a control chart of the data along with the tolerance bounds specified by side. plot.type = "hist" produces a histogram
of the data along with the tolerance bounds specified by side. plot.type = "both"
produces both the control chart and histogram. This argument is ignored when
plotting regression data.
plottol
69
x.lab
Specify the label for the x-axis.
y.lab
Specify the label for the y-axis.
z.lab
Specify the label for the z-axis.
...
Additional arguments passed to the plotting function used for the control charts
or regression scatterplots.
Value
plottol can return a control chart, histogram, or both for continuous data along with the calculated
tolerance intervals. For regression data, plottol returns a scatterplot along with the regression
tolerance bounds. For ANOVA output, plottol returns an interval plot for each factor.
References
Montgomery, D. C. (2005), Introduction to Statistical Quality Control, Fifth Edition, John Wiley &
Sons, Inc.
Examples
## 90%/90% 1-sided Weibull tolerance intervals for a sample
## of size 150.
set.seed(100)
x <- rweibull(150, 3, 75)
out <- exttol.int(x = x, alpha = 0.15, P = 0.90,
dist = "Weibull")
out
plottol(out, x, plot.type = "both", side = "lower",
x.lab = "Weibull Data")
## 90%/90% trivariate normal tolerance region.
set.seed(100)
x1 <- rnorm(100, 0, 0.2)
x2 <- rnorm(100, 0, 0.5)
x3 <- rnorm(100, 5, 1)
x <- cbind(x1, x2, x3)
mvtol.region(x = x, alpha = c(0.10, 0.05, 0.01),
P = c(0.90, 0.95, 0.99), B = 1000)
out2 <- mvtol.region(x = x, alpha = 0.10, P = 0.90, B = 1000)
out2
plottol(out2, x)
## 95%/95% 2-sided linear regression tolerance bounds
## for a sample of size 100.
set.seed(100)
x <- runif(100, 0, 10)
70
poislind.ll
y <- 20 + 5*x + rnorm(100, 0, 3)
out3 <- regtol.int(reg = lm(y ~ x), new.x = cbind(c(3, 6, 9)),
side = 2, alpha = 0.05, P = 0.95)
plottol(out3, x = cbind(1, x), y = y, side = "two", x.lab = "X",
y.lab = "Y")
poislind.ll
Maximum Likelihood Estimation for the Discrete Poisson-Lindley Distribution
Description
Performs maximum likelihood estimation for the parameter of the Poisson-Lindley distribution.
Usage
poislind.ll(x, theta = NULL, ...)
Arguments
x
A vector of raw data which is distributed according to a Poisson-Lindley distribution.
theta
Optional starting value for the parameter. If NULL, then the method of moments
estimator is used.
...
Additional arguments passed to the mle function.
Details
The discrete Poisson-Lindley distribution is a compound distribution that, potentially, provides a
better fit for count data relative to the traditional Poisson and negative binomial distributions.
Value
See the help file for mle to see how the output is structured.
References
Ghitany, M. E. and Al-Mutairi, D. K. (2009), Estimation Methods for the Discrete Poisson-Lindley
Distribution, Journal of Statistical Computation and Simulation, 79, 1–9.
Sankaran, M. (1970), The Discrete Poisson-Lindley Distribution, Biometrics, 26, 145–149.
See Also
mle, PoissonLindley
poislindtol.int
71
Examples
## Maximum likelihood estimation for randomly generated data
## from the Poisson-Lindley distribution.
set.seed(100)
pl.data <- rpoislind(n = 500, theta = 0.5)
out.pl <- poislind.ll(pl.data)
stats4::coef(out.pl)
stats4::vcov(out.pl)
poislindtol.int
Poisson-Lindley Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for data distributed according to the Poisson-Lindley
distribution.
Usage
poislindtol.int(x, m = NULL, alpha = 0.05, P = 0.99, side = 1,
...)
Arguments
x
A vector of raw data which is distributed according to a Poisson-Lindley distribution.
m
The number of observations in a future sample for which the tolerance limits
will be calculated. By default, m = NULL and, thus, m will be set equal to the
original sample size.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
...
Additional arguments passed to the poislind.ll function, which is used for
maximum likelihood estimation.
Details
The discrete Poisson-Lindley distribution is a compound distribution that, potentially, provides
a better fit for count data relative to the traditional Poisson and negative binomial distributions.
Poisson-Lindley distributions are heavily right-skewed distributions. For most practical applications, one will typically be interested in 1-sided upper bounds.
72
PoissonLindley
Value
poislindtol.int returns a data frame with the following items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
theta
MLE for the shape parameter theta.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Naghizadeh Qomi, M., Kiapour, A., and Young, D. S. (2015), Approximate Tolerance Intervals for
the Discrete Poisson-Lindley Distribution, Journal of Statistical Computation and Simulation, 86,
841–854.
See Also
PoissonLindley, poislind.ll
Examples
## 90%/90% 1-sided tolerance intervals for data assuming
## the Poisson-Lindley distribution.
x <- c(rep(0, 447), rep(1, 132), rep(2, 42), rep(3, 21),
rep(4, 3), rep(5, 2))
out <- poislindtol.int(x, alpha = 0.10, P = 0.90, side = 1)
out
PoissonLindley
Discrete Poisson-Lindley Distribution
Description
Density (mass), distribution function, quantile function, and random generation for the PoissonLindley distribution.
Usage
dpoislind(x,
ppoislind(q,
qpoislind(p,
rpoislind(n,
theta, log = FALSE)
theta, lower.tail = TRUE, log.p = FALSE)
theta, lower.tail = TRUE, log.p = FALSE)
theta)
PoissonLindley
73
Arguments
x, q
Vector of quantiles.
p
Vector of probabilities.
n
The number of observations. If length>1, then the length is taken to be the
number required.
theta
The shape parameter, which must be greater than 0.
log, log.p
Logical vectors. If TRUE, then the probabilities are given as log(p).
lower.tail
Logical vector. If TRUE, then probabilities are P [X ≤ x], else P [X > x].
Details
The Poisson-Lindley distribution has mass
p(x) =
θ2 (x + θ + 2)
,
(θ + 1)x+3
where x = 0, 1, . . . and θ > 0 is the shape parameter.
Value
dpoislind gives the density (mass), ppoislind gives the distribution function, qpoislind gives
the quantile function, and rpoislind generates random deviates for the specified distribution.
References
Ghitany, M. E. and Al-Mutairi, D. K. (2009), Estimation Methods for the Discrete Poisson-Lindley
Distribution, Journal of Statistical Computation and Simulation, 79, 1–9.
Sankaran, M. (1970), The Discrete Poisson-Lindley Distribution, Biometrics, 26, 145–149.
See Also
runif and .Random.seed about random number generation.
Examples
## Randomly generated data from the Poisson-Lindley
## distribution.
set.seed(100)
x <- rpoislind(n = 150, theta = 0.5)
hist(x, main = "Randomly Generated Data", prob = TRUE)
x.1 <- sort(x)
y <- dpoislind(x = x.1, theta = 0.5)
lines(x.1, y, col = 2, lwd = 2)
plot(x.1, ppoislind(q = x.1, theta = 0.5), type = "l",
xlab = "x", ylab = "Cumulative Probabilities")
74
poistol.int
qpoislind(p = 0.20, theta = 0.5, lower.tail = FALSE)
qpoislind(p = 0.80, theta = 0.5)
poistol.int
Poisson Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for Poisson random variables. From a statistical
quality control perspective, these limits bound the number of occurrences (which follow a Poisson
distribution) in a specified future time period.
Usage
poistol.int(x, n, m = NULL, alpha = 0.05, P = 0.99, side = 1,
method = c("TAB", "LS", "SC", "CC", "VS", "RVS",
"FT", "CSC"))
Arguments
x
The number of occurrences of the event in time period n. Can be a vector of
length n, in which case the sum of x is used.
n
The time period of the original measurements.
m
The specified future length of time. If m = NULL, then the tolerance limits will
be constructed assuming n for the future length of time.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of occurrences in future time lengths of size m to be covered by
this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
method
The method for calculating the lower and upper confidence bounds, which are
used in the calculation of the tolerance bounds. The default method is "TAB",
which is the tabular method and is usually preferred for a smaller number of
occurrences. "LS" gives the large-sample (Wald) method, which is usually preferred when the number of occurrences is x>20. "SC" gives the score method,
which again is usually used when the number of occurrences is relatively large.
"CC" gives a continuity-corrected version of the large-sample method. "VS"
gives a variance-stabilized version of the large-sample method. "RVS" is a recentered version of the variance-stabilization method. "FT" is the Freeman-Tukey
method. "CSC" is the continuity-corrected version of the score method. More
information on these methods can be found in the "References".
poistol.int
75
Value
poistol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of occurrences in future time periods of length m.
lambda.hat
The mean occurrence rate per unit time, calculated by x/n.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Barker, L. (2002), A Comparison of Nine Confidence Intervals for a Poisson Parameter When the
Expected Number of Events Is ≤ 5, The American Statistician, 56, 85–89.
Freeman, M. F. and Tukey, J. W. (1950), Transformations Related to the Angular and the Square
Root, Annals of Mathematical Statistics, 21, 607–611.
Hahn, G. J. and Chandra, R. (1981), Tolerance Intervals for Poisson and Binomial Variables, Journal of Quality Technology, 13, 100–110.
See Also
Poisson, umatol.int
Examples
## 95%/90% 1-sided Poisson tolerance limits for future
## occurrences in a period of length 3. All seven methods
## are presented for comparison.
poistol.int(x = 45, n
side = 1,
poistol.int(x = 45, n
side = 1,
poistol.int(x = 45, n
side = 1,
poistol.int(x = 45, n
side = 1,
poistol.int(x = 45, n
side = 1,
poistol.int(x = 45, n
side = 1,
poistol.int(x = 45, n
side = 1,
poistol.int(x = 45, n
side = 1,
= 9, m
method
= 9, m
method
= 9, m
method
= 9, m
method
= 9, m
method
= 9, m
method
= 9, m
method
= 9, m
method
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
3, alpha
"TAB")
3, alpha
"LS")
3, alpha
"SC")
3, alpha
"CC")
3, alpha
"VS")
3, alpha
"RVS")
3, alpha
"FT")
3, alpha
"CSC")
= 0.05, P = 0.90,
= 0.05, P = 0.90,
= 0.05, P = 0.90,
= 0.05, P = 0.90,
= 0.05, P = 0.90,
= 0.05, P = 0.90,
= 0.05, P = 0.90,
= 0.05, P = 0.90,
76
regtol.int
## 95%/90% 2-sided Poisson tolerance intervals for future
## occurrences in a period of length 15. All seven methods
## are presented for comparison.
poistol.int(x = 45, n
side = 2,
poistol.int(x = 45, n
side = 2,
poistol.int(x = 45, n
side = 2,
poistol.int(x = 45, n
side = 2,
poistol.int(x = 45, n
side = 2,
poistol.int(x = 45, n
side = 2,
poistol.int(x = 45, n
side = 2,
poistol.int(x = 45, n
side = 2,
regtol.int
= 9, m
method
= 9, m
method
= 9, m
method
= 9, m
method
= 9, m
method
= 9, m
method
= 9, m
method
= 9, m
method
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
15, alpha
"TAB")
15, alpha
"LS")
15, alpha
"SC")
15, alpha
"CC")
15, alpha
"VS")
15, alpha
"RVS")
15, alpha
"FT")
15, alpha
"CSC")
= 0.05, P = 0.90,
= 0.05, P = 0.90,
= 0.05, P = 0.90,
= 0.05, P = 0.90,
= 0.05, P = 0.90,
= 0.05, P = 0.90,
= 0.05, P = 0.90,
= 0.05, P = 0.90,
(Multiple) Linear Regression Tolerance Bounds
Description
Provides 1-sided or 2-sided (multiple) linear regression tolerance bounds. It is also possible to fit a
regression through the origin model.
Usage
regtol.int(reg, new.x = NULL, side = 1, alpha = 0.05, P = 0.99)
Arguments
reg
An object of class lm (i.e., the results from a linear regression routine).
new.x
Any new levels of the predictor(s) for which to report the tolerance bounds.
The number of columns must equal the number of predictors appearing on the
right-hand side of the model in reg.
side
Whether a 1-sided or 2-sided tolerance bound is required (determined by side = 1
or side = 2, respectively).
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by the tolerance bound(s).
regtol.int
77
Value
regtol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by the tolerance bound(s).
y
The value of the response given on the left-hand side of the model in reg.
y.hat
The predicted value of the response for the fitted linear regression model. This
data frame is sorted by this value.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Wallis, W. A. (1951), Tolerance Intervals for Linear Regression, in Second Berkeley Symposium on
Mathematical Statistics and Probability, ed. J. Neyman, Berkeley: University of CA Press, 43–51.
Young, D. S. (2013), Regression Tolerance Intervals, Communications in Statistics - Simulation and
Computation, 42, 2040–2055.
See Also
lm
Examples
## 95%/95% 2-sided linear regression tolerance bounds
## for a sample of size 100.
set.seed(100)
x <- runif(100, 0, 10)
y <- 20 + 5*x + rnorm(100, 0, 3)
out <- regtol.int(reg = lm(y ~ x), new.x = cbind(c(3, 6, 9)),
side = 2, alpha = 0.05, P = 0.95)
out
plottol(out, x = cbind(1, x), y = y, side = "two", x.lab = "X",
y.lab = "Y")
78
TwoParExponential
TwoParExponential
The 2-Parameter Exponential Distribution
Description
Density, distribution function, quantile function, and random generation for the 2-parameter exponential distribution with rate equal to rate and shift equal to shift.
Usage
d2exp(x,
p2exp(q,
q2exp(p,
r2exp(n,
rate
rate
rate
rate
=
=
=
=
1,
1,
1,
1,
shift
shift
shift
shift
=
=
=
=
0, log = FALSE)
0, lower.tail = TRUE, log.p = FALSE)
0, lower.tail = TRUE, log.p = FALSE)
0)
Arguments
x,q
Vector of quantiles.
p
Vector of probabilities.
n
The number of observations. If length>1, then the length is taken to be the
number required.
rate
Vector of rates.
shift
Vector of shifts.
log,log.p
Logical vectors. If TRUE, then probabilities are given as log(p).
lower.tail
Logical vector. If TRUE, then probabilities are P [X ≤ x], else P [X > x].
Details
If rate or shift are not specified, then they assume the default values of 1 and 0, respectively.
The 2-parameter exponential distribution has density
f (x) =
1 (x−µ)/β
e
β
where x ≥ µ, µ is the shift parameter, and β > 0 is the scale parameter.
Value
d2exp gives the density, p2exp gives the distribution function, q2exp gives the quantile function,
and r2exp generates random deviates.
See Also
runif and .Random.seed about random number generation.
umatol.int
79
Examples
## Randomly generated data from the 2-parameter exponential
## distribution.
set.seed(100)
x <- r2exp(n = 500, rate = 3, shift = -10)
hist(x, main = "Randomly Generated Data", prob = TRUE)
x.1 = sort(x)
y <- d2exp(x = x.1, rate = 3, shift = -10)
lines(x.1, y, col = 2, lwd = 2)
plot(x.1, p2exp(q = x.1, rate = 3, shift = -10), type = "l",
xlab = "x", ylab = "Cumulative Probabilities")
q2exp(p = 0.20, rate = 3, shift = -10, lower.tail = FALSE)
q2exp(p = 0.80, rate = 3, shift = -10)
umatol.int
Uniformly Most Accurate Upper Tolerance Limits for Certain Discrete
Distributions
Description
Provides uniformly most accurate upper tolerance limits for the binomial, negative binomial, and
Poisson distributions.
Usage
umatol.int(x, n = NULL, dist = c("Bin", "NegBin", "Pois"), N,
alpha = 0.05, P = 0.99)
Arguments
x
A vector of data which is distributed according to one of the binomial, negative
binomial, or Poisson distributions. If the length of x is 1, then it is assumed that
this number is the sum of iid values from the assumed distribution.
n
The sample size of the data. If null, then n is calculated as the length of x.
dist
The distribution for the data given by x. The options are "Bin" for the binomial
distribution, "NegBin" for the negative binomial distribution, and "Pois" for
the Poisson distribution.
N
Must be specified for the binomial and negative binomial distributions. If dist = "Bin",
then N is the number of Bernoulli trials and must be a positive integer. If
dist = "NegBin", then N is the total number of successful trials (or dispersion parameter) and must be strictly positive.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
80
uniftol.int
Value
umatol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
p.hat
The maximum likelihood estimate for the probability of success in each trial;
reported if dist = "Bin".
nu.hat
The maximum likelihood estimate for the probability of success in each trial;
reported if dist = "NegBin".
lambda.hat
The maximum likelihood estimate for the rate of success; reported if dist = "Pois".
1-sided.upper
The 1-sided upper tolerance limit.
References
Zacks, S. (1970), Uniformly Most Accurate Tolerance Limits for Monotone Likelihood Ratio Families of Discrete Distributions, Journal of the American Statistical Association, 65, 307–316.
See Also
Binomial, NegBinomial, Poisson
Examples
## Examples from Zacks (1970).
umatol.int(25,
P =
umatol.int(13,
P =
umatol.int(37,
uniftol.int
n = 4, dist = "Bin", N = 10, alpha = 0.10,
0.95)
n = 10, dist = "NegBin", N = 2, alpha = 0.10,
0.95)
n = 10, dist = "Pois", alpha = 0.10, P = 0.95)
Uniform Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for data distributed according to a uniform distribution.
Usage
uniftol.int(x, alpha = 0.05, P = 0.99, upper = NULL,
lower = NULL, side = 1)
uniftol.int
81
Arguments
x
A vector of data which is distributed according to a uniform distribution.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
upper
The upper bound of the data. When NULL, then the maximum of x is used.
lower
The lower bound of the data. When NULL, then the minimum of x is used.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
Value
uniftol.int returns a data frame with items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
References
Faulkenberry, G. D. and Weeks, D. L. (1968), Sample Size Determination for Tolerance Limits,
Technometrics, 10, 343–348.
Examples
## 90%/90% 1-sided uniform tolerance intervals for a sample
## of size 50 with a known lower bound of 0.
set.seed(100)
x <- runif(50, 0, 50)
out <- uniftol.int(x = x, alpha = 0.10, P = 0.90, lower = 0,
side = 1)
out
plottol(out, x, plot.type = "hist", side = "two",
x.lab = "Uniform Data")
82
ZipfMandelbrot
ZipfMandelbrot
Zipf-Mandelbrot Distributions
Description
Density (mass), distribution function, quantile function, and random generation for the Zipf, ZipfMandelbrot, and zeta distributions.
Usage
dzipfman(x, s,
pzipfman(q, s,
log.p
qzipfman(p, s,
log.p
rzipfman(n, s,
b
b
=
b
=
b
= NULL,
= NULL,
FALSE)
= NULL,
FALSE)
= NULL,
N = NULL, log = FALSE)
N = NULL, lower.tail = TRUE,
N = NULL, lower.tail = TRUE,
N = NULL)
Arguments
x, q
p
n
s, b
N
log, log.p
lower.tail
Vector of quantiles.
Vector of probabilities.
The number of observations. If length>1, then the length is taken to be the
number required.
The shape parameters, both of which must be greater than 0. b must be specified
for Zipf-Mandelbrot distributions.
The number of categories, which must be integer-valued for Zipf and ZipfMandelbrot distributions. For a zeta distribution, N = Inf must be used.
Logical vectors. If TRUE, then the probabilities are given as log(p).
Logical vector. If TRUE, then probabilities are P [X ≤ x], else P [X > x].
Details
The Zipf-Mandelbrot distribution has mass
(x + δ)−λ
p(x) = PN
,
−λ
i=1 (i + δ)
where x = 1, . . . , N , λ, δ > 0 are shape parameters, and N is the number of distinct categories. The
Zipf distribution is just a special case of the Zipf-Mandelbrot distribution where the second shape
parameter b=0. The zeta distribution has mass
p(x) =
x−λ
,
ζ(λ)
where x = 1, 2, . . ., λ > 1 is the shape parameter, and ζ() is the Riemann zeta function given by:
ζ(t) =
∞
X
1
< ∞.
t
i
i=1
ZipfMandelbrot
83
Note that the zeta distribution is just a special case of the Zipf distribution where s>1 and N goes to
infinity.
Value
dzipfman gives the density (mass), pzipfman gives the distribution function, qzipfman gives the
quantile function, and rzipfman generates random deviates for the specified distribution.
Note
These functions may be updated in a future version of the package so as to allow greater flexibility
with the inputs.
References
Mandelbrot, B. B. (1965), Information Theory and Psycholinguistics. In B. B. Wolman and E.
Nagel, editors. Scientific Psychology, Basic Books.
Young, D. S. (2013), Approximate Tolerance Limits for Zipf-Mandelbrot Distributions, Physica A:
Statistical Mechanics and its Applications, 392, 1702–1711.
Zipf, G. K. (1949), Human Behavior and the Principle of Least Effort, Hafner.
Zornig, P. and Altmann, G. (1995), Unified Representation of Zipf Distributions, Computational
Statistics and Data Analysis, 19, 461–473.
See Also
runif and .Random.seed about random number generation.
Examples
## Randomly generated data from the Zipf distribution.
set.seed(100)
x <- rzipfman(n = 150, s = 2, N = 100)
hist(x, main = "Randomly Generated Data", prob = TRUE)
x.1 <- sort(x)
y <- dzipfman(x = x.1, s = 2, N = 100)
lines(x.1, y, col = 2, lwd = 2)
plot(x.1, pzipfman(q = x.1, s = 2, N = 100), type = "l",
xlab = "x", ylab = "Cumulative Probabilities")
qzipfman(p = 0.20, s = 2, N = 100, lower.tail = FALSE)
qzipfman(p = 0.80, s = 2, N = 100)
## Randomly generated data from the Zipf-Mandelbrot distribution.
set.seed(100)
x <- rzipfman(n = 150, s = 2, b = 3, N = 100)
hist(x, main = "Randomly Generated Data", prob = TRUE)
84
zipftol.int
x.1 <- sort(x)
y <- dzipfman(x = x.1, s = 2, b = 3, N = 100)
lines(x.1, y, col = 2, lwd = 2)
plot(x.1, pzipfman(q = x.1, s = 2, b = 3, N = 100), type = "l",
xlab = "x", ylab = "Cumulative Probabilities")
qzipfman(p = 0.20, s = 2, b = 3, N = 100, lower.tail = FALSE)
qzipfman(p = 0.80, s = 2, b = 3, N = 100)
## Randomly generated data from the zeta distribution.
set.seed(100)
x <- rzipfman(n = 100, s = 1.3, N = Inf)
hist(x, main = "Randomly Generated Data", prob = TRUE)
x.1 <- sort(x)
y <- dzipfman(x = x.1, s = 1.3, N = Inf)
lines(x.1, y, col = 2, lwd = 2)
plot(x.1, pzipfman(q = x.1, s = 1.3, N = Inf), type = "l",
xlab = "x", ylab = "Cumulative Probabilities")
qzipfman(p = 0.20, s = 1.3, lower.tail = FALSE, N = Inf)
qzipfman(p = 0.80, s = 1.3, N = Inf)
zipftol.int
Zipf-Mandelbrot Tolerance Intervals
Description
Provides 1-sided or 2-sided tolerance intervals for data distributed according to Zipf, Zipf-Mandelbrot,
and zeta distributions.
Usage
zipftol.int(x, m = NULL, N = NULL, alpha = 0.05, P = 0.99,
side = 1, s = 1, b = 1, dist = c("Zipf",
"Zipf-Man", "Zeta"), ties = FALSE, ...)
Arguments
x
A vector of raw data or a table of counts which is distributed according to a Zipf,
Zipf-Mandelbrot, or zeta distribution. Do not supply a vector of counts!
m
The number of observations in a future sample for which the tolerance limits
will be calculated. By default, m = NULL and, thus, m will be set equal to the
original sample size.
zipftol.int
85
N
The number of categories when dist = "Zipf" or dist = "Zipf-Man". This
is not used when dist = "Zeta". If N = NULL, then N is estimated based on the
number of categories observed in the data.
alpha
The level chosen such that 1-alpha is the confidence level.
P
The proportion of the population to be covered by this tolerance interval.
side
Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1
or side = 2, respectively).
s
The initial value to estimate the shape parameter in the zm.ll function.
b
The initial value to estimate the second shape parameter in the zm.ll function
when dist = "Zipf-Man".
dist
Options are dist = "Zipf", dist = "Zipf-Man", or dist = "Zeta" if the
data is distributed according to the Zipf, Zipf-Mandelbrot, or zeta distribution,
respectively.
ties
How to handle if there are other categories with the same frequency as the category at the estimated tolerance limit. The default is FALSE, which does no
correction. If TRUE, then the highest ranked (i.e., lowest number) of the tied
categories is selected for the lower limit and the lowest ranked (i.e., highest
number) of the tied categories is selected for the upper limit.
...
Additional arguments passed to the zm.ll function, which is used for maximum
likelihood estimation.
Details
Zipf-Mandelbrot models are commonly used to model phenomena where the frequencies of categorical data are approximately inversely proportional to its rank in the frequency table. ZipfMandelbrot distributions are heavily right-skewed distributions with a (relatively) large mass placed
on the first category. For most practical applications, one will typically be interested in 1-sided upper bounds.
Value
zipftol.int returns a data frame with the following items:
alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
s.hat
MLE for the shape parameter s.
b.hat
MLE for the shape parameter b when dist = "Zipf-Man".
1-sided.lower
The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper
The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower
The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper
The 2-sided upper tolerance bound. This is given only if side = 2.
Note
This function may be updated in a future version of the package so as to allow greater flexibility
with the inputs.
86
zm.ll
References
Mandelbrot, B. B. (1965), Information Theory and Psycholinguistics. In B. B. Wolman and E.
Nagel, editors. Scientific Psychology, Basic Books.
Young, D. S. (2013), Approximate Tolerance Limits for Zipf-Mandelbrot Distributions, Physica A:
Statistical Mechanics and its Applications, 392, 1702–1711.
Zipf, G. K. (1949), Human Behavior and the Principle of Least Effort, Hafner.
Zornig, P. and Altmann, G. (1995), Unified Representation of Zipf Distributions, Computational
Statistics and Data Analysis, 19, 461–473.
See Also
ZipfMandelbrot, zm.ll
Examples
## 95%/99% 1-sided tolerance intervals for the Zipf,
## Zipf-Mandelbrot, and zeta distributions.
set.seed(100)
s <- 2
b <- 5
N <- 50
zipf.data <- rzipfman(n = 150, s = s, N = N)
zipfman.data <- rzipfman(n = 150, s = s, b = b, N = N)
zeta.data <- rzipfman(n = 150, s = s, N = Inf)
out.zipf <- zipftol.int(zipf.data, dist = "Zipf")
out.zipfman <- zipftol.int(zipfman.data, dist = "Zipf-Man")
out.zeta <- zipftol.int(zeta.data, N = Inf, dist = "Zeta")
out.zipf
out.zipfman
out.zeta
zm.ll
Maximum Likelihood Estimation for Zipf-Mandelbrot Models
Description
Performs maximum likelihood estimation for the parameters of the Zipf, Zipf-Mandelbrot, and zeta
distributions.
Usage
zm.ll(x, N = NULL, s = 1, b = 1, dist = c("Zipf", "Zipf-Man",
"Zeta"), ...)
zm.ll
87
Arguments
x
A vector of raw data or a table of counts which is distributed according to a Zipf,
Zipf-Mandelbrot, or zeta distribution. Do not supply a vector of counts!
N
The number of categories when dist = "Zipf" or dist = "Zipf-Man". This
is not used when dist = "Zeta". If N = NULL, then N is estimated based on the
number of categories observed in the data.
s
The initial value to estimate the shape parameter, which is set to 1 by default. If
a poor initial value is specified, then a WARNING message is returned.
b
The initial value to estimate the second shape parameter when dist = "Zipf-Man",
which is set to 1 by default. If a poor initial value is specified, then a WARNING
message is returned.
dist
Options are dist = "Zipf", dist = "Zipf-Man", or dist = "Zeta" if the
data is distributed according to the Zipf, Zipf-Mandelbrot, or zeta distribution,
respectively.
...
Additional arguments passed to the mle function.
Details
Zipf-Mandelbrot models are commonly used to model phenomena where the frequencies of categorical data are approximately inversely proportional to its rank in the frequency table.
Value
See the help file for mle to see how the output is structured.
Note
This function may be updated in a future version of the package so as to allow greater flexibility
with the inputs.
References
Mandelbrot, B. B. (1965), Information Theory and Psycholinguistics. In B. B. Wolman and E.
Nagel, editors. Scientific Psychology, Basic Books.
Zipf, G. K. (1949), Human Behavior and the Principle of Least Effort, Hafner.
Zornig, P. and Altmann, G. (1995), Unified Representation of Zipf Distributions, Computational
Statistics and Data Analysis, 19, 461–473.
See Also
mle, ZipfMandelbrot
88
zm.ll
Examples
## Maximum likelihood estimation for randomly generated data
## from the Zipf, Zipf-Mandelbrot, and zeta distributions.
set.seed(100)
s <- 2
b <- 5
N <- 50
zipf.data <- rzipfman(n = 500, s = s, N = N)
out.zipf <- zm.ll(zipf.data, N = N, dist = "Zipf")
stats4::coef(out.zipf)
stats4::vcov(out.zipf)
zipfman.data <- rzipfman(n = 500, s = s, b = b, N = N)
out.zipfman <- zm.ll(zipfman.data, N = N, dist = "Zipf-Man")
stats4::coef(out.zipfman)
diag(stats4::vcov(out.zipfman))
zeta.data <- rzipfman(n = 200, s = s, N = Inf)
out.zeta <- zm.ll(zeta.data, N = Inf, dist = "Zeta")
stats4::coef(out.zeta)
stats4::vcov(out.zeta)
Index
plottol, 68
poislind.ll, 70
poislindtol.int, 71
PoissonLindley, 72
poistol.int, 74
regtol.int, 76
TwoParExponential, 78
umatol.int, 79
uniftol.int, 80
ZipfMandelbrot, 82
zipftol.int, 84
zm.ll, 86
.Random.seed, 17, 19, 51, 73, 78, 83
∗Topic file
acc.samp, 4
anovatol.int, 5
bayesnormtol.int, 7
bintol.int, 9
bonftol.int, 12
cautol.int, 13
diffnormtol.int, 14
DiffProp, 16
DiscretePareto, 18
distfree.est, 19
dpareto.ll, 20
dparetotol.int, 21
exp2tol.int, 23
exptol.int, 25
exttol.int, 26
F1, 28
fidbintol.int, 29
fidnegbintol.int, 31
fidpoistol.int, 33
gamtol.int, 35
hypertol.int, 37
K.factor, 38
K.table, 40
laptol.int, 42
logistol.int, 44
mvregtol.region, 45
mvtol.region, 46
negbintol.int, 48
NegHypergeometric, 50
neghypertol.int, 52
nlregtol.int, 54
norm.OC, 55
norm.ss, 57
normtol.int, 59
np.order, 61
npregtol.int, 62
nptol.int, 64
paretotol.int, 66
acc.samp, 4, 38, 53
anova, 7
anovatol.int, 5
bayesnormtol.int, 7, 59
Binomial, 11, 80
bintol.int, 9
bonftol.int, 12
Cauchy, 14
cautol.int, 13
confint, 3
d2exp (TwoParExponential), 78
ddiffprop (DiffProp), 16
ddpareto (DiscretePareto), 18
diffnormtol.int, 14
DiffProp, 16, 28
DiscretePareto, 18, 21, 22
distfree.est, 19, 66
dnhyper (NegHypergeometric), 50
dpareto.ll, 20, 22
dparetotol.int, 21
dpoislind (PoissonLindley), 72
dzipfman (ZipfMandelbrot), 82
exp2tol.int, 23, 67
89
90
Exponential, 25
exptol.int, 25
exttol.int, 26
F1, 28
fidbintol.int, 29, 32, 34
fidnegbintol.int, 30, 31, 34
fidpoistol.int, 30, 32, 33
GammaDist, 36
gamtol.int, 35
Hypergeometric, 5, 38
hypertol.int, 37
integrate, 28, 40
K.factor, 7, 9, 15, 36, 38, 42, 57, 61
K.table, 40, 40
laptol.int, 42
lm, 7, 77
loess, 64
Logistic, 45
logistol.int, 44
mle, 21, 70, 87
mvregtol.region, 45
mvtol.region, 46
NegBinomial, 50, 80
negbintol.int, 48
NegHypergeometric, 50, 53
neghypertol.int, 52
nlregtol.int, 54
nls, 55
norm.OC, 55
norm.ss, 57
Normal, 9, 15, 59, 61
normtol.int, 7–9, 15, 40, 57, 59, 59
np.order, 61
npregtol.int, 62, 66
nptol.int, 20, 62–64, 64
p2exp (TwoParExponential), 78
paretotol.int, 66
pdiffprop (DiffProp), 16
pdpareto (DiscretePareto), 18
plottol, 68
pnhyper (NegHypergeometric), 50
INDEX
poislind.ll, 70, 72
poislindtol.int, 71
Poisson, 75, 80
PoissonLindley, 70, 72, 72
poistol.int, 74
ppoislind (PoissonLindley), 72
pzipfman (ZipfMandelbrot), 82
q2exp (TwoParExponential), 78
qdiffprop (DiffProp), 16
qdpareto (DiscretePareto), 18
qnhyper (NegHypergeometric), 50
qpoislind (PoissonLindley), 72
qzipfman (ZipfMandelbrot), 82
r2exp (TwoParExponential), 78
rdiffprop (DiffProp), 16
rdpareto (DiscretePareto), 18
regtol.int, 76
rnhyper (NegHypergeometric), 50
rpoislind (PoissonLindley), 72
runif, 17, 19, 51, 73, 78, 83
rzipfman (ZipfMandelbrot), 82
spline, 64
TDist, 39, 40
tolerance (tolerance-package), 3
tolerance-package, 3
TwoParExponential, 24, 67, 78
umatol.int, 11, 50, 75, 79
uniftol.int, 80
Weibull, 27
ZipfMandelbrot, 82, 86, 87
zipftol.int, 84
zm.ll, 86, 86